This document discusses Fourier series and integrals. It begins by explaining Fourier series using sines, cosines, and exponentials to represent periodic functions. Square waves are given as examples that can be expressed as infinite combinations of sines. Any periodic function can be expressed as a Fourier series. Fourier series are then derived for specific examples, including a square wave, repeating ramp, and up-down train of delta functions. Cosine series are also discussed. The document concludes by deriving the Fourier series for the delta function.
This document discusses iterative methods for solving systems of equations. It introduces the Jacobi iteration method and the Successive Over-Relaxation (SOR) method. SOR can accelerate the convergence compared to Jacobi by introducing an optimal relaxation parameter. Pseudocode is provided to implement SOR to iteratively solve a system of equations until the solution converges within a specified tolerance.
Solution to schrodinger equation with dirac comb potential slides
This document summarizes solving the Schrödinger equation for a Dirac comb potential. The potential is an infinite series of Dirac delta functions spaced periodically. Floquet theory is used to solve the time-independent Schrödinger equation for this potential. Boundary conditions are applied and the resulting equations are solved graphically. Allowed energy bands are determined and plotted versus wave vector for both attractive and repulsive delta function potentials.
William hyatt-7th-edition-drill-problems-solutionSalman Salman
This document contains solutions to drill problems from Chapter 2 on electrostatics. It includes calculations of electric fields, electric flux densities, and total charge for various charge distributions using Gauss's law and other concepts of electrostatics. Any errors found in the solutions should be reported to the author.
This document discusses techniques for calculating electric potential, including:
1. Laplace's equation and its solutions in 1D, 2D, and 3D, including boundary conditions.
2. The method of images, which uses fictitious "image" charges to solve problems involving conductors. The classical image problem and induced surface charge on a conductor are examined.
3. Other techniques like multipole expansion, separation of variables, and numerical methods like relaxation are mentioned but not explained in detail.
The document discusses vector spaces and related concepts:
1) It defines a vector space as a set V with vector addition and scalar multiplication operations that satisfy certain properties. Examples of vector spaces include R2, the plane in R3, and the space of real polynomials.
2) A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication and thus forms a vector space with the inherited operations. Examples given include the x-axis in Rn and solution spaces of linear differential equations.
3) The span of a set of vectors is the smallest subspace that contains those vectors, consisting of all possible linear combinations of the vectors in the set.
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
This document discusses iterative methods for solving systems of equations. It introduces the Jacobi iteration method and the Successive Over-Relaxation (SOR) method. SOR can accelerate the convergence compared to Jacobi by introducing an optimal relaxation parameter. Pseudocode is provided to implement SOR to iteratively solve a system of equations until the solution converges within a specified tolerance.
Solution to schrodinger equation with dirac comb potential slides
This document summarizes solving the Schrödinger equation for a Dirac comb potential. The potential is an infinite series of Dirac delta functions spaced periodically. Floquet theory is used to solve the time-independent Schrödinger equation for this potential. Boundary conditions are applied and the resulting equations are solved graphically. Allowed energy bands are determined and plotted versus wave vector for both attractive and repulsive delta function potentials.
William hyatt-7th-edition-drill-problems-solutionSalman Salman
This document contains solutions to drill problems from Chapter 2 on electrostatics. It includes calculations of electric fields, electric flux densities, and total charge for various charge distributions using Gauss's law and other concepts of electrostatics. Any errors found in the solutions should be reported to the author.
This document discusses techniques for calculating electric potential, including:
1. Laplace's equation and its solutions in 1D, 2D, and 3D, including boundary conditions.
2. The method of images, which uses fictitious "image" charges to solve problems involving conductors. The classical image problem and induced surface charge on a conductor are examined.
3. Other techniques like multipole expansion, separation of variables, and numerical methods like relaxation are mentioned but not explained in detail.
The document discusses vector spaces and related concepts:
1) It defines a vector space as a set V with vector addition and scalar multiplication operations that satisfy certain properties. Examples of vector spaces include R2, the plane in R3, and the space of real polynomials.
2) A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication and thus forms a vector space with the inherited operations. Examples given include the x-axis in Rn and solution spaces of linear differential equations.
3) The span of a set of vectors is the smallest subspace that contains those vectors, consisting of all possible linear combinations of the vectors in the set.
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
Linear Combination, Span And Linearly Independent, Dependent SetDhaval Shukla
In this presentation, the topic of Linear Combination, Span and Linearly Independent and Linearly Dependent Sets have been discussed. The sums for each topic have been given to understand the concept clearly for viewers.
The document discusses vector spaces and related linear algebra concepts. It defines vector spaces and lists the axioms that must be satisfied. Examples of vector spaces include the set of all pairs of real numbers and the space of 2x2 symmetric matrices. The document also discusses subspaces, linear combinations, span, basis, dimension, row space, column space, null space, rank, nullity, and change of basis. It provides examples and explanations of these fundamental linear algebra topics.
Monte Carlo methods for some not-quite-but-almost Bayesian problemsPierre Jacob
This document discusses Arthur Dempster's approach to statistical inference using Dempster-Shafer theory of belief functions. It involves sampling uniformly from the set Rx, which represents the feasible sets of parameter values θ that are consistent with the observed data. Rx can be represented using inequalities between ratios of components in u, related to the minimum path weights in a graph. This allows defining conditional distributions for components of u, enabling a Gibbs sampler to iteratively sample from the uniform distribution on Rx.
The document discusses inverse trigonometric functions and how to define their inverses by restricting the domains of the trig functions. It explains that the sine function's inverse is defined on [-1,1] and the cosine function's inverse is defined on [0,π]. Similarly, the tangent function's inverse is defined on (-π/2, π/2). Graphs and examples of the inverse sine, cosine, and tangent functions are provided.
This document provides notes on vector spaces, which are fundamental objects in linear algebra. It begins with examples of vector spaces such as R2, R3, C2, C3 and defines vector spaces more generally as sets that are closed under vector addition and scalar multiplication and satisfy other properties like the existence of additive identities. It then provides several examples of vector spaces including the set of all n-tuples over a field, the set of all m×n matrices, the set of differentiable functions on an interval, and the set of polynomials with coefficients in a field.
This document discusses the application of analytic functions to fluid flow, electrostatic fields, and heat flow problems. It explains that for incompressible fluid flow, the complex potential F(z) describes the flow, with its real part giving the velocity potential and imaginary part the stream function. It also describes how the electrostatic potential satisfies Laplace's equation and can be written as the real part of a complex potential. Finally, it explains that steady heat conduction problems are governed by Laplace's equation and the heat potential is the real part of a complex heat potential, with constant values representing isotherms and heat flow lines.
This document provides an overview of vector algebra concepts including scalars, vectors, unit vectors, position vectors, vector operations, gradients of scalars, divergence of vectors, and del operators. It defines these concepts, provides examples of their use in Cartesian, cylindrical and spherical coordinate systems, and works through examples of calculating gradients and divergences. The key topics covered are the definitions and calculations of gradients, divergences, and how to apply these concepts to vector fields in different coordinate systems.
The document defines a subspace as a non-empty subset W of a vector space V that is itself a vector space under the operations defined on V. It notes that every vector space has at least two subspaces: itself and the zero subspace containing only the zero vector. To prove that W is a subspace of V, we only need to verify that W is closed under the vector space operations. Examples are provided to illustrate this, such as showing that the set W={(x,0,0)| x in R} is a subspace of R3 by verifying it is closed under vector addition and scalar multiplication.
This document discusses key concepts in vector calculus including:
1) The gradient of a scalar, which is a vector representing the directional derivative/rate of change.
2) Divergence of a vector, which measures the outward flux density at a point.
3) Divergence theorem, relating the outward flux through a closed surface to the volume integral of the divergence.
4) Curl of a vector, which measures the maximum circulation and tendency for rotation.
Formulas are provided for calculating these quantities in Cartesian, cylindrical, and spherical coordinate systems. Examples are worked through applying the concepts and formulas.
This document provides information about vector spaces and subspaces. It defines a vector space as a set of objects called vectors that can be added together and multiplied by scalars, subject to certain rules. A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. The null space of a matrix is the set of solutions to the homogeneous equation Ax=0 and is a subspace. The column space of a matrix is the set of all linear combinations of its columns and is also a subspace. Examples are provided to illustrate these concepts.
Presentation of the work on Prime Numbers.
intended for mathematics loving people.
Please send comments and suggestions for improvement to solo.hermelin@gmail.com.
More presentations can be found in my website at http://solohermelin.com.
Deterministic finite automata (DFAs) are mathematical models of computation that can be used to represent regular languages. A DFA consists of: (1) a finite set of states, (2) a finite input alphabet, (3) a transition function mapping a state and input to another state, (4) a start state, and (5) a set of accepting states. DFAs can be represented visually using state transition diagrams or mathematically as a 5-tuple. Operations like union, intersection, and complement of languages can be modeled using operations like union, product, and complement of DFAs.
The document provides information about a linear algebra and vector calculus assignment for mechanical engineering students at L.D. College of Engineering in Ahmedabad, India. It includes the names of 10 students, an outline of 8 topics to be covered, and sample definitions, examples, and explanations related to those topics, such as definitions of vector spaces and subspaces, linear combinations, linear independence, and span of a set of vectors.
This document provides an introduction to z-transforms, which are a major mathematical tool for analyzing discrete systems including digital control systems. It discusses sequences and difference equations, which are foundational topics for understanding z-transforms. Specifically, it defines what a sequence is, distinguishes between first and second order difference equations, and explains how sequences can be shifted to the left or right. It also introduces key concepts like causal sequences and the unit step sequence that are important for z-transform analysis.
The document discusses Floquet theory applied to analyze the stability of internal gravity waves in a density-stratified fluid. It summarizes previous approaches to studying gravity wave instability, and outlines the author's thesis goal of using Floquet-Fourier computation to identify all physically unstable modes. This involves analyzing the Floquet exponents as a Riemann surface with complex wavenumber, to distinguish the two primary sheets corresponding to the physically relevant unstable solutions.
This document discusses linear transformations and their properties. It defines a linear transformation as a function between vector spaces that preserves vector addition and scalar multiplication. The kernel of a linear transformation is the set of vectors mapped to the zero vector, and is a subspace of the domain. The range is the set of images of all vectors under the transformation. Matrices can represent linear transformations, with the matrix equation representing the transformation of vectors. Examples are provided to illustrate key concepts such as kernels, ranges, and matrix representations of linear transformations.
The document discusses linear transformations between vector spaces. It defines key concepts such as the domain, codomain, range, and images/preimages of vectors under a linear transformation. A linear transformation preserves vector addition and scalar multiplication. It provides examples of linear transformations, such as rotations and projections in planes and spaces. It also discusses when a transformation defined by a matrix represents a linear transformation.
Differential equation & laplace transformation with matlabRavi Jindal
This document discusses using MATLAB to solve differential equations through Laplace transformations. It introduces key terms like the Laplace operator and generating function. It then demonstrates how to use MATLAB commands like "laplace" and "ilaplace" to calculate the Laplace transform of a function and take the inverse Laplace transform. Examples are provided, such as finding the Laplace transform of the function f(t)=-1.25+3.5t*exp(-2t)+1.25*exp(-2t).
The document discusses Legendre functions, which are solutions to Legendre's differential equation. Legendre functions arise when solving Laplace's equation in spherical coordinates. Legendre polynomials were first introduced by Adrien-Marie Legendre in 1785 as coefficients in an expansion of Newtonian potential. The document covers topics such as Legendre polynomials, Rodrigues' formula, orthogonality of Legendre polynomials, and associated Legendre functions.
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
The document discusses Fourier series and their applications. It begins by introducing how Fourier originally developed the technique to study heat transfer and how it can represent periodic functions as an infinite series of sine and cosine terms. It then provides the definition and examples of Fourier series representations. The key points are that Fourier series decompose a function into sinusoidal basis functions with coefficients determined by integrating the function against each basis function. The series may converge to the original function under certain conditions.
Linear Combination, Span And Linearly Independent, Dependent SetDhaval Shukla
In this presentation, the topic of Linear Combination, Span and Linearly Independent and Linearly Dependent Sets have been discussed. The sums for each topic have been given to understand the concept clearly for viewers.
The document discusses vector spaces and related linear algebra concepts. It defines vector spaces and lists the axioms that must be satisfied. Examples of vector spaces include the set of all pairs of real numbers and the space of 2x2 symmetric matrices. The document also discusses subspaces, linear combinations, span, basis, dimension, row space, column space, null space, rank, nullity, and change of basis. It provides examples and explanations of these fundamental linear algebra topics.
Monte Carlo methods for some not-quite-but-almost Bayesian problemsPierre Jacob
This document discusses Arthur Dempster's approach to statistical inference using Dempster-Shafer theory of belief functions. It involves sampling uniformly from the set Rx, which represents the feasible sets of parameter values θ that are consistent with the observed data. Rx can be represented using inequalities between ratios of components in u, related to the minimum path weights in a graph. This allows defining conditional distributions for components of u, enabling a Gibbs sampler to iteratively sample from the uniform distribution on Rx.
The document discusses inverse trigonometric functions and how to define their inverses by restricting the domains of the trig functions. It explains that the sine function's inverse is defined on [-1,1] and the cosine function's inverse is defined on [0,π]. Similarly, the tangent function's inverse is defined on (-π/2, π/2). Graphs and examples of the inverse sine, cosine, and tangent functions are provided.
This document provides notes on vector spaces, which are fundamental objects in linear algebra. It begins with examples of vector spaces such as R2, R3, C2, C3 and defines vector spaces more generally as sets that are closed under vector addition and scalar multiplication and satisfy other properties like the existence of additive identities. It then provides several examples of vector spaces including the set of all n-tuples over a field, the set of all m×n matrices, the set of differentiable functions on an interval, and the set of polynomials with coefficients in a field.
This document discusses the application of analytic functions to fluid flow, electrostatic fields, and heat flow problems. It explains that for incompressible fluid flow, the complex potential F(z) describes the flow, with its real part giving the velocity potential and imaginary part the stream function. It also describes how the electrostatic potential satisfies Laplace's equation and can be written as the real part of a complex potential. Finally, it explains that steady heat conduction problems are governed by Laplace's equation and the heat potential is the real part of a complex heat potential, with constant values representing isotherms and heat flow lines.
This document provides an overview of vector algebra concepts including scalars, vectors, unit vectors, position vectors, vector operations, gradients of scalars, divergence of vectors, and del operators. It defines these concepts, provides examples of their use in Cartesian, cylindrical and spherical coordinate systems, and works through examples of calculating gradients and divergences. The key topics covered are the definitions and calculations of gradients, divergences, and how to apply these concepts to vector fields in different coordinate systems.
The document defines a subspace as a non-empty subset W of a vector space V that is itself a vector space under the operations defined on V. It notes that every vector space has at least two subspaces: itself and the zero subspace containing only the zero vector. To prove that W is a subspace of V, we only need to verify that W is closed under the vector space operations. Examples are provided to illustrate this, such as showing that the set W={(x,0,0)| x in R} is a subspace of R3 by verifying it is closed under vector addition and scalar multiplication.
This document discusses key concepts in vector calculus including:
1) The gradient of a scalar, which is a vector representing the directional derivative/rate of change.
2) Divergence of a vector, which measures the outward flux density at a point.
3) Divergence theorem, relating the outward flux through a closed surface to the volume integral of the divergence.
4) Curl of a vector, which measures the maximum circulation and tendency for rotation.
Formulas are provided for calculating these quantities in Cartesian, cylindrical, and spherical coordinate systems. Examples are worked through applying the concepts and formulas.
This document provides information about vector spaces and subspaces. It defines a vector space as a set of objects called vectors that can be added together and multiplied by scalars, subject to certain rules. A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. The null space of a matrix is the set of solutions to the homogeneous equation Ax=0 and is a subspace. The column space of a matrix is the set of all linear combinations of its columns and is also a subspace. Examples are provided to illustrate these concepts.
Presentation of the work on Prime Numbers.
intended for mathematics loving people.
Please send comments and suggestions for improvement to solo.hermelin@gmail.com.
More presentations can be found in my website at http://solohermelin.com.
Deterministic finite automata (DFAs) are mathematical models of computation that can be used to represent regular languages. A DFA consists of: (1) a finite set of states, (2) a finite input alphabet, (3) a transition function mapping a state and input to another state, (4) a start state, and (5) a set of accepting states. DFAs can be represented visually using state transition diagrams or mathematically as a 5-tuple. Operations like union, intersection, and complement of languages can be modeled using operations like union, product, and complement of DFAs.
The document provides information about a linear algebra and vector calculus assignment for mechanical engineering students at L.D. College of Engineering in Ahmedabad, India. It includes the names of 10 students, an outline of 8 topics to be covered, and sample definitions, examples, and explanations related to those topics, such as definitions of vector spaces and subspaces, linear combinations, linear independence, and span of a set of vectors.
This document provides an introduction to z-transforms, which are a major mathematical tool for analyzing discrete systems including digital control systems. It discusses sequences and difference equations, which are foundational topics for understanding z-transforms. Specifically, it defines what a sequence is, distinguishes between first and second order difference equations, and explains how sequences can be shifted to the left or right. It also introduces key concepts like causal sequences and the unit step sequence that are important for z-transform analysis.
The document discusses Floquet theory applied to analyze the stability of internal gravity waves in a density-stratified fluid. It summarizes previous approaches to studying gravity wave instability, and outlines the author's thesis goal of using Floquet-Fourier computation to identify all physically unstable modes. This involves analyzing the Floquet exponents as a Riemann surface with complex wavenumber, to distinguish the two primary sheets corresponding to the physically relevant unstable solutions.
This document discusses linear transformations and their properties. It defines a linear transformation as a function between vector spaces that preserves vector addition and scalar multiplication. The kernel of a linear transformation is the set of vectors mapped to the zero vector, and is a subspace of the domain. The range is the set of images of all vectors under the transformation. Matrices can represent linear transformations, with the matrix equation representing the transformation of vectors. Examples are provided to illustrate key concepts such as kernels, ranges, and matrix representations of linear transformations.
The document discusses linear transformations between vector spaces. It defines key concepts such as the domain, codomain, range, and images/preimages of vectors under a linear transformation. A linear transformation preserves vector addition and scalar multiplication. It provides examples of linear transformations, such as rotations and projections in planes and spaces. It also discusses when a transformation defined by a matrix represents a linear transformation.
Differential equation & laplace transformation with matlabRavi Jindal
This document discusses using MATLAB to solve differential equations through Laplace transformations. It introduces key terms like the Laplace operator and generating function. It then demonstrates how to use MATLAB commands like "laplace" and "ilaplace" to calculate the Laplace transform of a function and take the inverse Laplace transform. Examples are provided, such as finding the Laplace transform of the function f(t)=-1.25+3.5t*exp(-2t)+1.25*exp(-2t).
The document discusses Legendre functions, which are solutions to Legendre's differential equation. Legendre functions arise when solving Laplace's equation in spherical coordinates. Legendre polynomials were first introduced by Adrien-Marie Legendre in 1785 as coefficients in an expansion of Newtonian potential. The document covers topics such as Legendre polynomials, Rodrigues' formula, orthogonality of Legendre polynomials, and associated Legendre functions.
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
The document discusses Fourier series and their applications. It begins by introducing how Fourier originally developed the technique to study heat transfer and how it can represent periodic functions as an infinite series of sine and cosine terms. It then provides the definition and examples of Fourier series representations. The key points are that Fourier series decompose a function into sinusoidal basis functions with coefficients determined by integrating the function against each basis function. The series may converge to the original function under certain conditions.
This document discusses power series solutions to differential equations, specifically Bessel's equations. It provides background on power series expansions and their properties. It explains that solutions to differential equations can be written as power series when the coefficients of the equation are analytic at a point. As an example, it finds the general solution to a second order differential equation using the power series method. In summary, it outlines techniques for solving differential equations using power series expansions at ordinary points.
This document discusses applying power series solutions of Bessel equations to solve problems involving struts with variable moments of inertia. It begins by reviewing the basics of power series solutions to differential equations and Bessel equations. It then develops a model of the variable strut problem as a Bessel-form differential equation. Finally, it solves this equation using the power series method for Bessel equations to obtain a result for problems involving struts with non-constant moments of inertia.
This document discusses various topics related to integration including indefinite and definite integrals, power rules, properties of integrals, integration by parts, u-substitution, and definitions. Some key points covered are:
- Integration was developed independently by Newton and Leibniz in the late 17th century.
- The indefinite integral finds an antiderivative, while the definite integral evaluates the area under a function between bounds.
- Common integration techniques include power rules, integration by parts, and u-substitution.
- Integration rules and properties allow integrals to be transformed and simplified.
The document discusses quantum mechanical concepts including:
1) The time derivative of the momentum expectation value satisfies an equation involving the potential gradient.
2) For an infinite potential well, the kinetic energy expectation value is proportional to n^2/a^2 and the potential energy expectation value vanishes.
3) Eigenfunctions of an eigenvalue problem under certain boundary conditions correspond to positive eigenvalues that are sums of squares of integer multiples of pi.
This document discusses integrals involving exponential functions. It shows that integrating the exponential function results in dividing the constant in the exponent. It evaluates the important definite integral from 0 to infinity of e^-ax, which equals 1/a. It also evaluates the double integral from -infinity to infinity of e^-a(x^2+y^2), which equals sqrt(pi/a). Taking derivatives of these integrals generates related integrals involving x and x^4 that are useful in kinetic theory of gases.
This document discusses conjugate gradient methods for minimizing quadratic functions. It begins by introducing quadratic functions and noting that conjugate gradient methods can minimize them without needing the full Hessian matrix, unlike Newton's method. It then defines what it means for a set of vectors to be conjugate with respect to a positive definite matrix A. Vectors are conjugate if their inner products with respect to A are all zero. The document proves that a set of conjugate vectors forms a basis and describes a simple conjugate gradient algorithm that finds the minimum in n iterations using n conjugate search directions.
Contour integrals provide a useful technique for evaluating certain integrals. This document introduces contour integrals and provides examples to illustrate their application. It explains that contour integrals can be viewed as line integrals and that Cauchy's theorem allows continuous deformation of the contour without changing the result, as long as it does not cross singularities. The key points are:
1) Contour integrals around a pole give the residue of the pole times 2πi.
2) Examples are worked out to show contour integrals give the same results as conventional integrals for integrals that cannot be expressed in terms of elementary functions.
3) Contour integrals enable evaluation of new integrals that would be difficult to evaluate otherwise, like calculating an integral of an exponential
This document contains exercises related to dynamical systems and periodic points. It includes the following summaries:
1. The doubling map on the circle has 2n-1 periodic points of period n. Its periodic points are dense.
2. The map f(x)=|x-2| has a fixed point at x=1. Other periodic and pre-periodic points are [0,2]\{1\} of period 2 and (-∞,0)∪(2,+∞) which are pre-periodic.
3. Expanding maps of the circle are topologically mixing since intervals get longer under iteration, eventually covering the entire circle.
The document discusses complex eigenvalues and eigenvectors for systems of linear differential equations. It shows that if the matrix A has complex conjugate eigenvalue pairs r1 and r2, then the corresponding eigenvectors and solutions will also be complex conjugates. This leads to real-valued fundamental solutions that can express the general solution. An example demonstrates these concepts, finding the complex eigenvalues and eigenvectors and expressing the general solution in terms of real-valued functions. Spiral points, centers, eigenvalues, and trajectory behaviors are also summarized.
(1) The document discusses various integration techniques including: review of integral formulas, integration by parts, trigonometric integrals involving products of sines and cosines, trigonometric substitutions, and integration of rational functions using partial fractions.
(2) Examples are provided to demonstrate each technique, such as using integration by parts to evaluate integrals of the form ∫udv, using trigonometric identities to reduce powers of trigonometric functions, and using partial fractions to break down rational functions into simpler fractions.
(3) The key techniques discussed are integration by parts, trigonometric substitutions to transform integrals involving quadratic expressions into simpler forms, and partial fractions to decompose rational functions for integration. Various examples illustrate the
7.curves Further Mathematics Zimbabwe Zimsec Cambridgealproelearning
The document discusses properties of curves defined by functions. It begins by listing objectives for understanding important points on graphs like maxima, minima, and inflection points. It emphasizes using graphing technology to experiment but not substitute for analytical work. Examples are provided to demonstrate finding maximums, minimums, intersections, and asymptotes of various functions. The key points are determining features of a curve from its defining function.
1) The document defines and discusses the domains and ranges of inverse trigonometric functions such as sin-1x, cos-1x, and tan-1x.
2) The inverse functions are defined based on reflecting portions of the original trigonometric functions over the line y=x.
3) The domains and ranges of the inverse functions are restricted to ensure each inverse function is a single-valued function.
The document discusses Taylor series and how they can be used to approximate functions. It provides an example of using Taylor series to approximate the cosine function. Specifically:
1) It derives the Taylor series for the cosine function centered at x=0.
2) It shows that this Taylor series converges absolutely for all x.
3) It demonstrates that the Taylor series equals the cosine function everywhere based on properties of the remainder term.
4) It provides an example of using the Taylor series to approximate cos(0.1) to within 10^-7, the accuracy of a calculator display.
On the Seidel’s Method, a Stronger Contraction Fixed Point Iterative Method o...BRNSS Publication Hub
In the solution of a system of linear equations, there exist many methods most of which are not fixed point iterative methods. However, this method of Sidel’s iteration ensures that the given system of the equation must be contractive after satisfying diagonal dominance. The theory behind this was discussed in sections one and two and the end; the application was extensively discussed in the last section.
This document summarizes a method for efficiently generating fractal images using a multi-grid approach. It views the process of generating fractals via iterated function systems as a Markov process. This allows modeling the image as the stationary distribution of the Markov process. Calculating the stationary distribution requires solving a large set of linear equations defined by the state transition matrix. The multi-grid method is adapted for this problem by using the affine transformations that define the iterated function system to interpolate between grid levels and perform smoothing. This replaces traditional interpolation and smoothing steps and exploits the specific structure of the problem to provide a more efficient solution than standard multi-grid approaches.
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This document provides information about statistics concepts including measures of central tendency (mean, median, mode), calculating mean and median for grouped and ungrouped data, frequency distributions, and ogives. It also includes 50 multiple choice questions testing understanding of these statistical concepts. Key topics covered are calculating and comparing means, medians, and modes, determining class boundaries and mid-values, and identifying appropriate measures and formulas for grouped and ungrouped data sets.
This document discusses probability and provides examples of calculating probability. It covers key concepts like experimental probability, theoretical probability, mutually exclusive events, complementary events, exhaustive events, and sure events. It then provides 30 multiple choice questions testing understanding of probability concepts like finding the probability of drawing a particular ball or card from a set.
This document discusses applications of trigonometry, including determining the height or length of objects using angles of elevation/depression and trigonometric ratios. It provides examples like calculating the height of a pole given the observation angle and distance, or finding the width of a river based on the boat's angle and distance traveled. The document ends with 20 multiple choice questions testing these trigonometric application concepts.
The document provides information about mensuration and calculating volumes and surface areas of different geometric solids like cubes, spheres, cylinders, cones, and prisms. It includes definitions of these shapes, formulas to calculate their measurements, sample problems, and a practice test with multiple choice questions related to volumes and surface areas of combinations of solids.
1. The document discusses various properties of tangents and secants to circles, including: a secant intersects a circle in two points, a tangent intersects in one point, and a tangent is perpendicular to the radius at the point of contact.
2. It provides examples of lengths of tangents from internal and external points and how tangents from an external point are equal in length.
3. The document also covers areas and lengths of sectors of circles based on the central angle subtended, as well as properties of common tangents between two circles.
This document provides information about similar triangles over 5 pages. It begins by defining similar figures and triangles, and explaining the properties of similar triangles including equal corresponding angles and proportional corresponding sides. It then lists various criteria and properties to determine if triangles are similar, such as AAA, SSS, and angle-side criteria. The document concludes with 50 multiple choice questions related to similar triangles.
This document provides information about coordinate geometry including:
1. The definitions of the abscissa and ordinate of a point and the coordinates of points on the x-axis and y-axis.
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1. CHAPTER 4
FOURIER SERIES AND INTEGRALS
4.1
FOURIER SERIES FOR PERIODIC FUNCTIONS
This section explains three Fourier series: sines, cosines, and exponentials eikx.
Square waves (1 or 0 or −1) are great examples, with delta functions in the derivative.
We look at a spike, a step function, and a ramp—and smoother functions too.
Start with sin x. It has period 2π since sin(x + 2π) = sin x. It is an odd function
since sin(−x) = − sin x, and it vanishes at x = 0 and x = π. Every function sin nx
has those three properties, and Fourier looked at infinite combinations of the sines:
∞
Fourier sine series
S(x) = b1 sin x + b2 sin 2x + b3 sin 3x + · · · =
bn sin nx (1)
n=1
If the numbers b1 , b2 , . . . drop off quickly enough (we are foreshadowing the importance of the decay rate) then the sum S(x) will inherit all three properties:
Periodic S(x + 2π) = S(x)
Odd S(−x) = −S(x)
S(0) = S(π) = 0
200 years ago, Fourier startled the mathematicians in France by suggesting that any
function S(x) with those properties could be expressed as an infinite series of sines.
This idea started an enormous development of Fourier series. Our first step is to
compute from S(x) the number bk that multiplies sin kx.
Suppose S(x) =
bn sin nx. Multiply both sides by sin kx. Integrate from 0 to π:
π
π
S(x) sin kx dx =
0
π
b1 sin x sin kx dx + · · · +
0
bk sin kx sin kx dx + · · · (2)
0
On the right side, all integrals are zero except the highlighted one with n = k.
This property of “orthogonality” will dominate the whole chapter. The sines make
90◦ angles in function space, when their inner products are integrals from 0 to π:
π
Orthogonality
sin nx sin kx dx = 0 if n = k .
0
317
(3)
2. 318
Chapter 4 Fourier Series and Integrals
sin mx π
m
0
cos mx dx =
Zero comes quickly if we integrate
= 0 − 0. So we use this:
1
1
cos(n − k)x − cos(n + k)x .
(4)
2
2
Integrating cos mx with m = n − k and m = n + k proves orthogonality of the sines.
Product of sines
sin nx sin kx =
The exception is when n = k. Then we are integrating (sin kx)2 =
π
π
sin kx sin kx dx =
0
0
1
dx −
2
π
0
1
2
− 1 cos 2kx:
2
π
1
cos 2kx dx = .
2
2
(5)
The highlighted term in equation (2) is bkπ/2. Multiply both sides of (2) by 2/π:
Sine coefficients
S(−x) = −S(x)
bk =
2
π
π
S(x) sin kx dx =
0
1 π
S(x) sin kx dx.
π −π
(6)
Notice that S(x) sin kx is even (equal integrals from −π to 0 and from 0 to π).
I will go immediately to the most important example of a Fourier sine series. S(x)
is an odd square wave with SW (x) = 1 for 0 < x < π. It is drawn in Figure 4.1 as
an odd function (with period 2π) that vanishes at x = 0 and x = π.
SW (x) = 1
−π
π
0
2π
E
x
Figure 4.1: The odd square wave with SW (x + 2π) = SW (x) = {1 or 0 or −1}.
Example 1 Find the Fourier sine coefficients bk of the square wave SW (x).
Solution
For k = 1, 2, . . . use the first formula (6) with S(x) = 1 between 0 and π:
bk =
2
π
π
sin kx dx =
0
2 − cos kx
π
k
π
=
0
2
π
2 0 2 0 2 0
, , , , , ,...
1 2 3 4 5 6
(7)
The even-numbered coefficients b2k are all zero because cos 2kπ = cos 0 = 1. The
odd-numbered coefficients bk = 4/πk decrease at the rate 1/k. We will see that same
1/k decay rate for all functions formed from smooth pieces and jumps.
Put those coefficients 4/πk and zero into the Fourier sine series for SW (x):
Square wave
SW (x) =
4 sin x sin 3x sin 5x sin 7x
+
+
+
+···
π
1
3
5
7
(8)
Figure 4.2 graphs this sum after one term, then two terms, and then five terms. You
can see the all-important Gibbs phenomenon appearing as these “partial sums”
3. 4.1 Fourier Series for Periodic Functions
319
include more terms. Away from the jumps, we safely approach SW (x) = 1 or −1.
At x = π/2, the series gives a beautiful alternating formula for the number π:
1=
4 1 1 1 1
− + − +···
π 1 3 5 7
so that
π=4
1
1
−
1
3
+
1
5
−
1
7
+··· .
(9)
The Gibbs phenomenon is the overshoot that moves closer and closer to the jumps.
Its height approaches 1.18 . . . and it does not decrease with more terms of the series!
Overshoot is the one greatest obstacle to calculation of all discontinuous functions
(like shock waves in fluid flow). We try hard to avoid Gibbs but sometimes we can’t.
4 sin x sin 3x
+
π
1
3
4 sin x
Dashed
π 1
−π
π
4 sin x
sin 9x
+···+
π
1
9
overshoot−→
SW = 1
5 terms:
Solid curve
x
x
π
2
Figure 4.2: Gibbs phenomenon: Partial sums
N
1
bn sin nx overshoot near jumps.
Fourier Coefficients are Best
Let me look again at the first term b1 sin x = (4/π) sin x. This is the closest possible
approximation to the square wave SW , by any multiple of sin x (closest in the least
squares sense). To see this optimal property of the Fourier coefficients, minimize the
error over all b1 :
π
π
(SW −b1 sin x)2 dx The b1 derivative is −2
The error is
0
(SW −b1 sin x) sin x dx.
0
The integral of sin2 x is π/2. So the derivative is zero when b1 = (2/π)
This is exactly equation (6) for the Fourier coefficient.
π
0
S(x) sin x dx.
Each bk sin kx is as close as possible to SW (x). We can find the coefficients bk
one at a time, because the sines are orthogonal. The square wave has b2 = 0 because
all other multiples of sin 2x increase the error. Term by term, we are “projecting the
function onto each axis sin kx.”
Fourier Cosine Series
The cosine series applies to even functions with C(−x) = C(x):
∞
Cosine series C(x) = a0 + a1 cos x + a2 cos 2x + · · · = a0 +
an cos nx.
n=1
(10)
4. 320
Chapter 4 Fourier Series and Integrals
Every cosine has period 2π. Figure 4.3 shows two even functions, the repeating
ramp RR(x) and the up-down train UD(x) of delta functions. That sawtooth
ramp RR is the integral of the square wave. The delta functions in UD give the
derivative of the square wave. (For sines, the integral and derivative are cosines.)
RR and UD will be valuable examples, one smoother than SW , one less smooth.
First we find formulas for the cosine coefficients a0 and ak . The constant term a0
is the average value of the function C(x):
a0 = Average
a0 =
π
1
π
C(x) dx =
0
π
1
C(x) dx.
2π −π
(11)
I just integrated every term in the cosine series (10) from 0 to π. On the right side,
the integral of a0 is a0 π (divide both sides by π). All other integrals are zero:
π
0
sin nx
cos nx dx =
n
π
= 0 − 0 = 0.
(12)
0
In words, the constant function 1 is orthogonal to cos nx over the interval [0, π].
The other cosine coefficients ak come from the orthogonality of cosines. As with
sines, we multiply both sides of (10) by cos kx and integrate from 0 to π:
π
π
C(x) cos kx dx =
0
π
a0 cos kx dx+
0
π
ak(cos kx)2 dx+··
a1 cos x cos kx dx+··+
0
0
You know what is coming. On the right side, only the highlighted term can be
nonzero. Problem 4.1.1 proves this by an identity for cos nx cos kx—now (4) has a
plus sign. The bold nonzero term is akπ/2 and we multiply both sides by 2/π:
Cosine coefficients
C(−x) = C(x)
ak =
2
π
π
C(x) cos kx dx =
0
1 π
C(x) cos kx dx .
π −π
(13)
Again the integral over a full period from −π to π (also 0 to 2π) is just doubled.
2δ(x) T
RR(x) = |x|
−π
0
π
E
2π
Repeating Ramp RR(x)
Integral of Square Wave
x
2δ(x − 2π) T
Up-down U D(x)
−π
0
−2δ(x + π)
c
π
2π
E
x
−2δ(x − π)
c
Figure 4.3: The repeating ramp RR and the up-down UD (periodic spikes) are even.
The derivative of RR is the odd square wave SW . The derivative of SW is U D.
5. 4.1 Fourier Series for Periodic Functions
321
Example 2 Find the cosine coefficients of the ramp RR(x) and the up-down UD(x).
Solution
The simplest way is to start with the sine series for the square wave:
SW (x) =
4 sin x sin 3x sin 5x sin 7x
+
+
+
+··· .
π
1
3
5
7
Take the derivative of every term to produce cosines in the up-down delta function:
Up-down series
UD(x) =
4
[cos x + cos 3x + cos 5x + cos 7x + · · · ] .
π
(14)
Those coefficients don’t decay at all. The terms in the series don’t approach zero, so
officially the series cannot converge. Nevertheless it is somehow correct and important.
Unofficially this sum of cosines has all 1’s at x = 0 and all −1’s at x = π. Then +∞
and −∞ are consistent with 2δ(x) and −2δ(x − π). The true way to recognize δ(x) is
by the test δ(x)f (x) dx = f (0) and Example 3 will do this.
For the repeating ramp, we integrate the square wave series for SW (x) and add the
average ramp height a0 = π/2, halfway from 0 to π:
Ramp series RR(x) =
π π cos x cos 3x cos 5x cos 7x
−
+
+
+
+··· .
2
4 12
32
52
72
(15)
The constant of integration is a0 . Those coefficients ak drop off like 1/k 2 . They could be
computed directly from formula (13) using x cos kx dx, but this requires an integration
by parts (or a table of integrals or an appeal to Mathematica or Maple). It was much
easier to integrate every sine separately in SW (x), which makes clear the crucial point:
Each “degree of smoothness” in the function is reflected in a faster decay rate of its
Fourier coefficients ak and bk .
No decay
1/k decay
1/k2 decay
1/k4 decay
r k decay with r < 1
Delta functions (with spikes)
Step functions (with jumps)
Ramp functions (with corners)
Spline functions (jumps in f )
Analytic functions like 1/(2 − cos x)
Each integration divides the kth coefficient by k. So the decay rate has an extra
1/k. The “Riemann-Lebesgue lemma” says that ak and bk approach zero for any
continuous function (in fact whenever |f (x)|dx is finite). Analytic functions achieve
a new level of smoothness—they can be differentiated forever. Their Fourier series
and Taylor series in Chapter 5 converge exponentially fast.
The poles of 1/(2 − cos x) will be complex solutions of cos x = 2. Its Fourier series
converges quickly because r k decays faster than any power 1/k p . Analytic functions
are ideal for computations—the Gibbs phenomenon will never appear.
Now we go back to δ(x) for what could be the most important example of all.
6. 322
Chapter 4 Fourier Series and Integrals
Example 3 Find the (cosine) coefficients of the delta function δ(x), made 2π-periodic.
Solution The spike occurs at the start of the interval [0, π] so safer to integrate from
−π to π. We find a0 = 1/2π and the other ak = 1/π (cosines because δ(x) is even):
Average a0 =
π
1
1
δ(x) dx =
2π −π
2π
Cosines ak =
1
1 π
δ(x) cos kx dx =
π −π
π
Then the series for the delta function has all cosines in equal amounts:
Delta function
δ(x) =
1
1
+ [cos x + cos 2x + cos 3x + · · · ] .
2π π
(16)
Again this series cannot truly converge (its terms don’t approach zero). But we can graph
the sum after cos 5x and after cos 10x. Figure 4.4 shows how these “partial sums” are
doing their best to approach δ(x). They oscillate faster and faster away from x = 0.
Actually there is a neat formula for the partial sum δN (x) that stops at cos Nx. Start
by writing each term 2 cos θ as eiθ + e−iθ :
δN =
1
1
[1 + 2 cos x + · · · + 2 cos Nx] =
1 + eix + e−ix + · · · + eiN x + e−iN x .
2π
2π
This is a geometric progression that starts from e−iN x and ends at eiN x . We have powers
of the same factor eix . The sum of a geometric series is known:
Partial sum
up to cos N x
1
δN (x) =
1
1 ei(N + 2 )x − e−i(N + 2 )x
1 sin(N + 1 )x
2
.
=
2π
eix/2 − e−ix/2
2π
sin 1 x
2
(17)
This is the function graphed in Figure 4.4. We claim that for any N the area underneath
δN (x) is 1. (Each cosine integrated from −π to π gives zero. The integral of 1/2π is
1.) The central “lobe” in the graph ends when sin(N + 1 )x comes down to zero, and
2
that happens when (N + 1 )x = ±π. I think the area under that lobe (marked by bullets)
2
approaches the same number 1.18 . . . that appears in the Gibbs phenomenon.
In what way does δN (x) approach δ(x)? The terms cos nx in the series jump around
1
at each point x = 0, not approaching zero. At x = π we see 2π [1 − 2 + 2 − 2 + · · · ] and
the sum is 1/2π or −1/2π. The bumps in the partial sums don’t get smaller than 1/2π.
The right test for the delta function δ(x) is to multiply by a smooth f (x) =
ak cos kx
and integrate, because we only know δ(x) from its integrals δ(x)f (x) dx = f (0):
Weak convergence
of δN (x) to δ(x)
π
−π
δN (x)f (x) dx = a0 + · · · + aN → f (0) .
(18)
In this integrated sense (weak sense) the sums δN (x) do approach the delta function !
The convergence of a0 + · · · + aN is the statement that at x = 0 the Fourier series of a
smooth f (x) =
ak cos kx converges to the number f (0).
7. 4.1 Fourier Series for Periodic Functions
δ10 (x)
height 21/2π
δ5 (x)
−π
323
height 11/2π
height 1/2π
π height −1/2π
0
Figure 4.4: The sums δN (x) = (1 + 2 cos x + · · · + 2 cos Nx)/2π try to approach δ(x).
Complete Series: Sines and Cosines
Over the half-period [0, π], the sines are not orthogonal to all the cosines. In fact the
integral of sin x times 1 is not zero. So for functions F (x) that are not odd or even,
we move to the complete series (sines plus cosines) on the full interval. Since our
functions are periodic, that “full interval” can be [−π, π] or [0, 2π]:
∞
Complete Fourier series
F (x) = a0 +
∞
an cos nx +
n=1
bn sin nx .
(19)
n=1
On every “2π interval” all sines and cosines are mutually orthogonal. We find the
Fourier coefficients ak and bk in the usual way: Multiply (19) by 1 and cos kx and
sin kx, and integrate both sides from −π to π:
a0 =
π
1
1 π
1 π
F (x) dx ak =
F (x) cos kx dx bk =
F (x) sin kx dx. (20)
2π −π
π −π
π −π
Orthogonality kills off infinitely many integrals and leaves only the one we want.
Another approach is to split F (x) = C(x) + S(x) into an even part and an odd
part. Then we can use the earlier cosine and sine formulas. The two parts are
C(x) = Feven (x) =
F (x) + F (−x)
2
S(x) = Fodd (x) =
F (x) − F (−x)
.
2
(21)
The even part gives the a’s and the odd part gives the b’s. Test on a short square
pulse from x = 0 to x = h—this one-sided function is not odd or even.
8. 324
Chapter 4 Fourier Series and Integrals
1 for 0 < x < h
0 for h < x < 2π
Example 4 Find the a’s and b’s if F (x) = square pulse =
Solution The integrals for a0 and ak and bk stop at x = h where F (x) drops to zero.
The coefficients decay like 1/k because of the jump at x = 0 and the drop at x = h:
Coefficients of square pulse
ak =
1
π
h
cos kx dx =
a0 =
sin kh
bk =
πk
0
1
2π
h
1 dx =
0
1
π
h
= average
2π
h
sin kx dx =
0
1 − cos kh
πk
.
(22)
1
If we divide F (x) by h, its graph is a tall thin rectangle: height h , base h, and area = 1.
When h approaches zero, F (x)/h is squeezed into a very thin interval. The tall
rectangle approaches (weakly) the delta function δ(x). The average height is area/2π =
1/2π. Its other coefficients ak /h and bk /h approach 1/π and 0, already known for δ(x):
F (x)
→ δ(x)
h
1 sin kh
1
ak
=
→
h
π kh
π
and
bk
1 − cos kh
=
→ 0 as h → 0. (23)
h
πkh
When the function has a jump, its Fourier series picks the halfway point. This
example would converge to F (0) = 1 and F (h) = 1 , halfway up and halfway down.
2
2
The Fourier series converges to F (x) at each point where the function is smooth.
This is a highly developed theory, and Carleson won the 2006 Abel Prize by proving
convergence for every x except a set of measure zero. If the function has finite energy
|F (x)|2 dx, he showed that the Fourier series converges “almost everywhere.”
Energy in Function = Energy in Coefficients
There is an extremely important equation (the energy identity) that comes from
integrating (F (x))2 . When we square the Fourier series of F (x), and integrate from
−π to π, all the “cross terms” drop out. The only nonzero integrals come from 12
and cos2 kx and sin2 kx, multiplied by a2 and a2 and b2 :
0
k
k
Energy in F (x) =
π
(F (x))2dx
−π
π
−π
(a0 +
ak cos kx +
bk sin kx)2 dx
= 2πa2 + π(a2 + b2 + a2 + b2 + · · · ).
0
1
1
2
2
(24)
The energy in F (x) equals the energy in the coefficients. The left side is like the
length squared of a vector, except the vector is a function. The right side comes from
an infinitely long vector of a’s and b’s. The lengths are equal, which says that the
Fourier transform from√
function to vector is like an orthogonal matrix. Normalized
√
by constants 2π and π, we have an orthonormal basis in function space.
What is this function space ? It is like ordinary 3-dimensional space, except the
“vectors” are functions. Their length f comes from integrating instead of adding:
f 2 = |f (x)|2 dx. These functions fill Hilbert space. The rules of geometry hold:
10. 326
Chapter 4 Fourier Series and Integrals
Notice that c0 = a0 is still the average of F (x), because e0 = 1. The orthogonality
of einx and eikx is checked by integrating, as always. But the complex inner product
(F, G) takes the complex conjugate G of G. Before integrating, change eikx to e−ikx :
Complex inner product
π
(F, G) =
−π
F (x)G(x) dx
Orthogonality of einx and eikx
π
π
ei(n−k)x
ei(n−k)x dx =
= 0.
i(n − k) −π
−π
(27)
Example 5 Add the complex series for 1/(2 − eix ) and 1/(2 − e−ix ). These geometric
series have exponentially fast decay from 1/2k . The functions are analytic.
1 eix e2ix
+
+
+ ·· +
2
4
8
1 e−ix e−2ix
+
+
+ ··
2
4
8
=1+
cos x cos 2x cos 3x
+
+
+ ··
2
4
8
When we add those functions, we get a real analytic function:
4 − 2 cos x
1
(2 − e−ix ) + (2 − eix )
1
=
+
=
ix
−ix
ix )(2 − e−ix )
2−e
2−e
(2 − e
5 − 4 cos x
(28)
This ratio is the infinitely smooth function whose cosine coefficients are 1/2 k .
Example 6 Find ck for the 2π-periodic shifted pulse F (x) =
Solution
1 for s ≤ x ≤ s + h
0 elsewhere in [−π, π]
The integrals (26) from −π to π become integrals from s to s + h:
s+h
s+h
1
1 e−ikx
ck =
1 · e−ikx dx =
2π s
2π −ik s
= e−iks
1 − e−ikh
2πik
.
(29)
Notice above all the simple effect of the shift by s. It “modulates” each c k by e−iks . The
energy is unchanged, the integral of |F |2 just shifts, and all |e−iks | = 1:
Shift F (x) to F (x − s) ←→ Multiply ck by e−iks .
(30)
Example 7 Centered pulse with shift s = −h/2. The square pulse becomes centered
around x = 0. This even function equals 1 on the interval from −h/2 to h/2:
Centered by s = − h
2
ck = eikh/2
1 sin(kh/2)
1 − e−ikh
=
.
2πik
2π
k/2
Divide by h for a tall pulse. The ratio of sin(kh/2) to kh/2 is the sinc function:
Tall pulse
1
Fcentered
=
h
2π
∞
sinc
−∞
kh
2
eikx =
1/h for − h/2 ≤ x ≤ h/2
0
elsewhere in [−π, π]
1
That division by h produces area = 1. Every coefficient approaches 2π as h → 0.
The Fourier series for the tall thin pulse again approaches the Fourier series for δ(x).
11. 327
4.1 Fourier Series for Periodic Functions
Hilbert space can contain vectors c = (c0 , c1 , c−1 , c2 , c−2 , · · · ) instead of functions
F (x). The length of c is 2π |ck |2 = |F |2dx. The function space is often denoted
by L2 and the vector space is 2 . The energy identity is trivial (but deep). Integrating
the Fourier series for F (x) times F (x), orthogonality kills every cn ck for n = k. This
leaves the ck ck = |ck |2 :
π
−π
|F (x)|2dx =
π
(
−π
cn einx )(
ck e−ikx )dx = 2π(|c0 |2 + |c1 |2 + |c−1 |2 + ··) . (31)
This is Plancherel’s identity: The energy in x-space equals the energy in k-space.
Finally I want to emphasize the three big rules for operating on F (x) =
1.
2.
3.
ck eikx :
dF
has Fourier coefficients ikck (energy moves to high k).
dx
ck
The integral of F (x) has Fourier coefficients , k = 0 (faster decay).
ik
The derivative
The shift to F (x−s) has Fourier coefficients e−iks ck (no change in energy).
Application: Laplace’s Equation in a Circle
Our first application is to Laplace’s equation. The idea is to construct u(x, y) as an
infinite series, choosing its coefficients to match u0 (x, y) along the boundary. Everything depends on the shape of the boundary, and we take a circle of radius 1.
Begin with the simple solutions 1, r cos θ, r sin θ, r 2 cos 2θ, r 2 sin 2θ, ... to Laplace’s
equation. Combinations of these special solutions give all solutions in the circle:
u(r, θ) = a0 + a1 r cos θ + b1 r sin θ + a2 r 2 cos 2θ + b2 r 2 sin 2θ + · · ·
(32)
It remains to choose the constants ak and bk to make u = u0 on the boundary.
For a circle u0 (θ) is periodic, since θ and θ + 2π give the same point:
Set r = 1
u0 (θ) = a0 + a1 cos θ + b1 sin θ + a2 cos 2θ + b2 sin 2θ + · · ·
(33)
This is exactly the Fourier series for u0 . The constants ak and bk must be the
Fourier coefficients of u0 (θ). Thus the problem is completely solved, if an infinite
series (32) is acceptable as the solution.
Example 8 Point source u0 = δ(θ) at θ = 0 The whole boundary is held at u 0 = 0,
except for the source at x = 1, y = 0. Find the temperature u(r, θ) inside.
Fourier series for δ
u0 (θ) =
1
1
1
+ (cos θ + cos 2θ + cos 3θ + · · · ) =
2π π
2π
∞
einθ
−∞
12. 328
Chapter 4 Fourier Series and Integrals
Inside the circle, each cos nθ is multiplied by r n :
Infinite series for u
u(r, θ) =
1
1
+ (r cos θ + r 2 cos 2θ + r 3 cos 3θ + · · · ) (34)
2π π
Poisson managed to sum this infinite series! It involves a series of powers of re iθ .
So we know the response at every (r, θ) to the point source at r = 1, θ = 0:
Temperature inside circle
u(r, θ) =
1
1 − r2
2π 1 + r 2 − 2r cos θ
(35)
At the center r = 0, this produces the average of u0 = δ(θ) which is a0 = 1/2π. On the
boundary r = 1, this produces u = 0 except at the point source where cos 0 = 1:
On the ray θ = 0
u(r, θ) =
1 1+r
1
1 − r2
=
.
2π 1 + r 2 − 2r
2π 1 − r
(36)
As r approaches 1, the solution becomes infinite as the point source requires.
Example 9 Solve for any boundary values u0 (θ) by integrating over point sources.
When the point source swings around to angle ϕ, the solution (35) changes from θ to
θ − ϕ. Integrate this “Green’s function” to solve in the circle:
Poisson’s formula
u(r, θ) =
1
2π
π
−π
u0 (ϕ)
1 − r2
dϕ
1 + r 2 − 2r cos(θ − ϕ)
(37)
Ar r = 0 the fraction disappears and u is the average u0 (ϕ)dϕ/2π. The steady
state temperature at the center is the average temperature around the circle.
Poisson’s formula illustrates a key idea. Think of any u 0(θ) as a circle of point sources.
The source at angle ϕ = θ produces the solution inside the integral (37). Integrating
around the circle adds up the responses to all sources and gives the response to u 0 (θ).
Example 10 u0 (θ) = 1 on the top half of the circle and u0 = −1 on the bottom half.
Solution
The boundary values are the square wave SW (θ). Its sine series is in (8):
Square wave for u0 (θ)
SW (θ) =
4 sin θ sin 3θ sin 5θ
+
+
+···
π
1
3
5
(38)
Inside the circle, multiplying by r, r 2 , r 3 ,... gives fast decay of high frequencies:
Rapid decay inside
u(r, θ) =
4 r sin θ r 3 sin 3θ r 5 sin 5θ
+
+
+···
π
1
3
5
Laplace’s equation has smooth solutions, even when u0 (θ) is not smooth.
(39)
13. 4.1 Fourier Series for Periodic Functions
329
WORKED EXAMPLE
A hot metal bar is moved into a freezer (zero temperature). The sides of the bar
are coated so that heat only escapes at the ends. What is the temperature u(x, t)
along the bar at time t? It will approach u = 0 as all the heat leaves the bar.
Solution The heat equation is ut = uxx . At t = 0 the whole bar is at a constant
temperature, say u = 1. The ends of the bar are at zero temperature for all time t 0.
This is an initial-boundary value problem:
Heat equation
ut = uxx
with u(x, 0) = 1 and u(0, t) = u(π, t) = 0. (40)
Those zero boundary conditions suggest a sine series. Its coefficients depend on t:
∞
Series solution of the heat equation
u(x, t) =
bn (t) sin nx.
(41)
1
The form of the solution shows separation of variables. In a comment below, we
look for products A(x) B(t) that solve the heat equation and the boundary conditions.
What we reach is exactly A(x) = sin nx and the series solution (41).
Two steps remain. First, choose each bn (t) sin nx to satisfy the heat equation:
Substitute into ut = uxx
bn (t) sin nx = −n2 bn (t) sin nx
2
bn (t) = e−n t bn (0).
Notice bn = −n2 bn . Now determine each bn (0) from the initial condition u(x, 0) = 1
on (0, π). Those numbers are the Fourier sine coefficients of SW (x) in equation (38):
∞
Box function/square wave
bn (0) sin nx = 1
bn (0) =
1
4
for odd n
πn
This completes the series solution of the initial-boundary value problem:
Bar temperature
u(x, t) =
odd
4 −n2 t
e
sin nx.
πn
n
(42)
2
For large n (high frequencies) the decay of e−n t is very fast. The dominant term
(4/π)e−t sin x for large times will come from n = 1. This is typical of the heat
equation and all diffusion, that the solution (the temperature profile) becomes very
smooth as t increases.
Numerical difficulty I regret any bad news in such a beautiful solution. To compute
u(x, t), we would probably truncate the series in (42) to N terms. When that finite
series is graphed on the website, serious bumps appear in uN (x, t). You ask if there
is a physical reason but there isn’t. The solution should have maximum temperature
at the midpoint x = π/2, and decay smoothly to zero at the ends of the bar.
14. 330
Chapter 4 Fourier Series and Integrals
Those unphysical bumps are precisely the Gibbs phenomenon. The initial
u(x, 0) is 1 on (0, π) but its odd reflection is −1 on (−π, 0). That jump has produced
the slow 4/πn decay of the coefficients, with Gibbs oscillations near x = 0 and x = π.
The sine series for u(x, t) is not a success numerically. Would finite differences help?
Separation of variables We found bn (t) as the coefficient of an eigenfunction sin nx.
Another good approach is to put u = A(x) B(t) directly into ut = uxx :
Separation
A(x) B (t) = A (x) B(t) requires
B (t)
A (x)
=
= constant. (43)
A(x)
B(t)
A /A is constant in space, B /B is constant in time, and they are equal:
√
√
A
= −λ gives A = sin λ x and cos λ x
A
B
= −λ gives B = e−λt
B
√
√
The products AB = e−λt sin λ x and e−λt cos λ x solve the heat equation for any
number λ. But the boundary condition u(0, t) =√0 eliminates the cosines. Then
u(π, t) = 0 requires λ = n2 = 1, 4, 9, . . . to have sin λ π = 0. Separation of variables
has recovered the functions in the series solution (42).
Finally u(x, 0) = 1 determines the numbers 4/πn for odd n. We find zero for even
n because sin nx has n/2 positive loops and n/2 negative loops. For odd n, the extra
positive loop is a fraction 1/n of all loops, giving slow decay of the coefficients.
Heat bath (the opposite problem) The solution on the website is 1 − u(x, t),
because it solves a different problem. The bar is initially frozen at U(x, 0) =
0. It is placed into a heat bath at the fixed temperature U = 1 (or U = T0 ).
The new unknown is U and its boundary conditions are no longer zero.
The heat equation and its boundary conditions are solved first by UB (x, t). In
this example UB ≡ 1 is constant. Then the difference V = U − UB has zero boundary
values, and its initial values are V = −1. Now the eigenfunction method (or separation of variables) solves for V . (The series in (42) is multiplied by −1 to account
for V (x, 0) = −1.) Adding back UB solves the heat bath problem: U = UB + V =
1 − u(x, t).
Here UB ≡ 1 is the steady state solution at t = ∞, and V is the transient solution.
The transient starts at V = −1 and decays quickly to V = 0.
Heat bath at one end The website problem is different in another way too. The
Dirichlet condition u(π, t) = 1 is replaced by the Neumann condition u (1, t) = 0.
Only the left end is in the heat bath. Heat flows down the metal bar and out at the
far end, now located at x = 1. How does the solution change for fixed-free?
Again UB = 1 is a steady state. The boundary conditions apply to V = 1 − UB :
Fixed-free
eigenfunctions
V (0) = 0 and V (1) = 0 lead to A(x) = sin n +
1
2
πx. (44)
15. 4.1 Fourier Series for Periodic Functions
331
Those eigenfunctions give a new form for the sum of Bn (t) An (x):
Fixed-free solution
1 2 2
π t
Bn (0) e−(n+ 2 )
V (x, t) =
sin n +
odd n
1
πx.
2
(45)
All frequencies shift by 1 and multiply by π, because A = −λA has a free end
2
at x = 1. The crucial question is: Does orthogonality still hold for these new
eigenfunctions sin n + 1 πx on [0, 1]? The answer is yes because this fixed-free
2
“Sturm–Liouville problem” A = −λA is still symmetric.
Summary The series solutions all succeed but the truncated series all fail. We can
see the overall behavior of u(x, t) and V (x, t). But their exact values close to the
jumps are not computed well until we improve on Gibbs.
We could have solved the fixed-free problem on [0, 1] with the fixed-fixed solution
on [0, 2]. That solution will be symmetric around x = 1 so its slope there is zero.
Then rescaling x by 2π changes sin(n + 1 )πx into sin(2n + 1)x. I hope you like the
2
graphics created by Aslan Kasimov on the cse website.
Problem Set 4.1
1
Find the Fourier series on −π ≤ x ≤ π for
(a)
(b)
(c)
(d)
f (x) = sin3 x, an odd function
f (x) = | sin x|, an even function
f (x) = x
f (x) = ex , using the complex form of the series.
What are the even and odd parts of f (x) = ex and f (x) = eix ?
2
From Parseval’s formula the square wave sine coefficients satisfy
π(b2 + b2 + · · · ) =
1
2
π
−π
|f (x)|2 dx =
Derive the remarkable sum π 2 = 8(1 +
3
1
9
+
1
25
π
1 dx = 2π.
−π
+ · · · ).
If a square pulse is centered at x = 0 to give
π
π
, f (x) = 0 for
|x| π,
2
2
draw its graph and find its Fourier coefficients ak and bk .
f (x) = 1 for |x|
4
Suppose f has period T instead of 2x, so that f (x) = f (x + T ). Its graph from
−T /2 to T /2 is repeated on each successive interval and its real and complex
Fourier series are
∞
2πx
2πx
+ b1 sin
+··· =
ck eik2πx/T
f (x) = a0 + a1 cos
T
T
−∞
Multiplying by the right functions and integrating from −T /2 to T /2, find ak ,
bk , and ck .
16. 332
5
Chapter 4 Fourier Series and Integrals
Plot the first three partial sums and the function itself:
x(π − x) =
8
π
sin x sin 3x sin 5x
+
+
+···
1
27
125
, 0 x π.
Why is 1/k 3 the decay rate for this function? What is the second derivative?
6
What constant function is closest in the least square sense to f = cos2 x? What
multiple of cos x is closest to f = cos3 x?
7
Sketch the 2π-periodic half wave with f (x) = sin x for 0 x π and f (x) = 0
for −π x 0. Find its Fourier series.
8
(a) Find the lengths of the vectors u = (1, 1 , 1 , 1 , . . .) and v = (1, 1 , 1 , . . .) in
2 4 8
3 9
Hilbert space and test the Schwarz inequality |uT v|2 ≤ (uT u)(v T v).
(b) For the functions f = 1 + 1 eix + 1 e2ix + · · · and g = 1 + 1 eix + 1 e2ix + · · ·
2
4
3
9
use part (a) to find the numerical value of each term in
2
π
f (x) g(x) dx
−π
≤
π
−π
|f (x)|2 dx
π
−π
|g(x)|2 dx.
Substitute for f and g and use orthogonality (or Parseval).
9
Find the solution to Laplace’s equation with u0 = θ on the boundary. Why is
this the imaginary part of 2(z − z 2 /2 + z 3 /3 · · · ) = 2 log(1 + z)? Confirm that
on the unit circle z = eiθ , the imaginary part of 2 log(1 + z) agrees with θ.
10
If the boundary condition for Laplace’s equation is u0 = 1 for 0 θ π and
u0 = 0 for −π θ 0, find the Fourier series solution u(r, θ) inside the unit
circle. What is u at the origin?
11
With boundary values u0 (θ) = 1 + 1 eiθ + 1 e2iθ + · · · , what is the Fourier series
2
4
solution to Laplace’s equation in the circle? Sum the series.
12
(a) Verify that the fraction in Poisson’s formula satisfies Laplace’s equation.
(b) What is the response u(r, θ) to an impulse at the point (0, 1), at the angle
ϕ = π/2?
(c) If u0 (ϕ) = 1 in the quarter-circle 0 ϕ π/2 and u0 = 0 elsewhere, show
that at points on the horizontal axis (and especially at the origin)
u(r, 0) =
1
1
+
tan−1
2 2π
1 − r2
−2r
1
dϕ
=√
tan−1
2 − c2
b + c cos ϕ
b
√
by using
b2 − c2 sin ϕ
c + b cos ϕ
.
17. 4.1 Fourier Series for Periodic Functions
13
333
When the centered square pulse in Example 7 has width h = π, find
(a) its energy
|F (x)|2 dx by direct integration
(b) its Fourier coefficients ck as specific numbers
(c) the sum in the energy identity (31) or (24)
If h = 2π, why is c0 = 1 the only nonzero coefficient ? What is F (x)?
14
In Example 5, F (x) = 1 + (cos x)/2 + · · ·+ (cos nx)/2n +· · · is infinitely smooth:
(a) If you take 10 derivatives, what is the Fourier series of d10 F/dx10 ?
(b) Does that series still converge quickly? Compare n10 with 2n for n1024 .
15
(A touch of complex analysis) The analytic function in Example 5 blows up
when 4 cos x = 5. This cannot happen for real x, but equation (28) shows
1
blowup if eix = 2 or 2 . In that case we have poles at x = ±i log 2. Why are
there also poles at all the complex numbers x = ±i log 2 + 2πn ?
16
(A second touch) Change 2’s to 3’s so that equation (28) has 1/(3 − eix ) +
1/(3 − e−ix ). Complete that equation to find the function that gives fast decay
at the rate 1/3k .
17
(For complex professors only) Change those 2’s and 3’s to 1’s:
2 − eix − e−ix
1
(1 − e−ix ) + (1 − eix )
1
=
+
=
= 1.
1 − eix 1 − e−ix
(1 − eix )(1 − e−ix )
2 − eix − e−ix
A constant ! What happened to the pole at eix = 1 ? Where is the dangerous
series (1 + eix + · · · ) + (1 + e−ix + · · · ) = 2 + 2 cos x + · · · involving δ(x) ?
18
Following the Worked Example, solve the heat equation ut = uxx from a point
source u(x, 0) = δ(x) with free boundary conditions u (π, t) = u (−π, t) = 0.
Use the infinite cosine series for δ(x) with time decay factors bn (t).