Bessel functions are solutions to Bessel's differential equation and describe oscillations that arise in many physical systems. Friedrich Bessel first systematically analyzed solutions to this equation in 1824, which became known as Bessel functions. There are Bessel functions of the first kind (Jp(x)) and second kind (Yp(x)). Jp(x) is bounded at x=0 while Yp(x) is unbounded, making them linearly independent solutions for the general solution. The gamma function was developed to define Bessel functions for all real values of p.
The document discusses partial differentiation and its applications. It covers functions of two variables, first and second partial derivatives, and applications including the Cobb-Douglas production function and finding marginal productivity from a production function. Examples are provided to demonstrate calculating partial derivatives of various functions and applying partial derivatives in contexts like production analysis.
Bessel's equation describes functions that arise in various physical problems, such as vibrating membranes and radar. The equation can be solved using an extended power series method to derive Bessel functions of the first kind, which are characterized by their orthogonality properties and represent solutions as a sum of integer powers of x.
This ppt covers the topic of B.Sc.1 Mathematics,unit - 5 , paper - 2, calculus- Introduction of Linear differential equation of second order , complete solution in terms of known integral belonging to the complementary function.
- A differential equation relates an independent variable, dependent variable, and derivatives of the dependent variable with respect to the independent variable.
- The order of a differential equation is the order of the highest derivative, and the degree is the degree of the highest derivative.
- Differential equations can be classified based on their order (first order vs higher order) and linearity (linear vs nonlinear).
- The general solution of a differential equation contains arbitrary constants, while a particular solution gives specific values for those constants.
The document discusses partial differential equations (PDEs). It defines PDEs and gives their general form involving independent variables, dependent variables, and partial derivatives. It describes methods for obtaining the complete integral, particular solution, singular solution, and general solution of a PDE. It provides examples of types of PDEs and how to solve them by assuming certain forms for the dependent and independent variables and their partial derivatives.
B.tech ii unit-2 material beta gamma functionRai University
1. The document discusses the gamma and beta functions, which are defined in terms of improper definite integrals involving exponential and power functions.
2. Examples are provided to demonstrate properties and applications of the gamma function, including evaluating integrals involving the gamma function.
3. The beta function is defined in terms of an integral from 0 to 1, and its relationship to the gamma function is described.
1) A differential equation contains an independent variable (x), a dependent variable (y), and the derivative of the dependent variable with respect to the independent variable (dy/dx).
2) The order of a differential equation refers to the highest order derivative present. For example, an equation containing dy/dx would be first order, while one containing d2y/dx2 would be second order.
3) The degree of a differential equation refers to the highest power of the highest order derivative. For example, an equation containing (d2y/dx) would be degree 1, while one containing (d2y/dx)2 would be degree 2.
4) There are several methods for solving first
The document discusses partial differentiation and its applications. It covers functions of two variables, first and second partial derivatives, and applications including the Cobb-Douglas production function and finding marginal productivity from a production function. Examples are provided to demonstrate calculating partial derivatives of various functions and applying partial derivatives in contexts like production analysis.
Bessel's equation describes functions that arise in various physical problems, such as vibrating membranes and radar. The equation can be solved using an extended power series method to derive Bessel functions of the first kind, which are characterized by their orthogonality properties and represent solutions as a sum of integer powers of x.
This ppt covers the topic of B.Sc.1 Mathematics,unit - 5 , paper - 2, calculus- Introduction of Linear differential equation of second order , complete solution in terms of known integral belonging to the complementary function.
- A differential equation relates an independent variable, dependent variable, and derivatives of the dependent variable with respect to the independent variable.
- The order of a differential equation is the order of the highest derivative, and the degree is the degree of the highest derivative.
- Differential equations can be classified based on their order (first order vs higher order) and linearity (linear vs nonlinear).
- The general solution of a differential equation contains arbitrary constants, while a particular solution gives specific values for those constants.
The document discusses partial differential equations (PDEs). It defines PDEs and gives their general form involving independent variables, dependent variables, and partial derivatives. It describes methods for obtaining the complete integral, particular solution, singular solution, and general solution of a PDE. It provides examples of types of PDEs and how to solve them by assuming certain forms for the dependent and independent variables and their partial derivatives.
B.tech ii unit-2 material beta gamma functionRai University
1. The document discusses the gamma and beta functions, which are defined in terms of improper definite integrals involving exponential and power functions.
2. Examples are provided to demonstrate properties and applications of the gamma function, including evaluating integrals involving the gamma function.
3. The beta function is defined in terms of an integral from 0 to 1, and its relationship to the gamma function is described.
1) A differential equation contains an independent variable (x), a dependent variable (y), and the derivative of the dependent variable with respect to the independent variable (dy/dx).
2) The order of a differential equation refers to the highest order derivative present. For example, an equation containing dy/dx would be first order, while one containing d2y/dx2 would be second order.
3) The degree of a differential equation refers to the highest power of the highest order derivative. For example, an equation containing (d2y/dx) would be degree 1, while one containing (d2y/dx)2 would be degree 2.
4) There are several methods for solving first
This document provides an introduction to ordinary differential equations (ODEs). It defines ODEs as differential equations containing functions of one independent variable and its derivatives. The document discusses some key concepts related to ODEs including order, degree, and different types of ODEs such as variable separable, homogeneous, exact, linear, and Bernoulli's equations. Examples of each type of ODE are provided along with the general methods for solving each type.
Second order homogeneous linear differential equations Viraj Patel
1) The document discusses second order linear homogeneous differential equations, which have the general form P(x)y'' + Q(x)y' + R(x)y = 0.
2) It describes methods for finding the general solution including reduction of order, and discusses the solutions when the coefficients are constants.
3) The general solution depends on the nature of the roots of the auxiliary equation: distinct real roots, repeated real roots, or complex roots.
Partial differential equation & its application.isratzerin6
Partial differential equations (PDEs) involve partial derivatives of dependent variables with respect to more than one independent variable. PDEs can be linear if the dependent variable and all its partial derivatives occur linearly, or non-linear. PDEs are used to model systems in fields like physics, engineering, and quantum mechanics, with examples being the Laplace, heat, and wave equations used in fluid dynamics, heat transfer, and quantum mechanics respectively. The heat equation specifically describes the distribution of heat over time in a given region.
The document discusses Fourier series and two of their applications. Fourier series can be used to represent periodic functions as an infinite series of sines and cosines. This allows approximating functions that are not smooth using trigonometric polynomials. Two key applications are representing forced oscillations, where a periodic driving force can be modeled as a Fourier series, and solving the heat equation, where the method of separation of variables results in a Fourier series representation of temperature over space and time.
The document discusses complex differentiability and analytic functions. It shows that for a function f(z) to be complex differentiable, the Cauchy-Riemann equations relating the partial derivatives of the real and imaginary parts must be satisfied. It also discusses representing functions as power series and their radii of convergence. Multivalued functions like logarithms and roots are discussed, noting the need for branch cuts to define single-valued branches.
The document discusses Bessel functions, which are solutions to a second order differential equation that arises in diverse situations. It also discusses the Hankel transform, which expresses functions as a weighted sum of Bessel functions of the first kind. The Hankel transform is useful for problems with cylindrical or spherical symmetry, as it appears when taking the multidimensional Fourier transform in hyperspherical coordinates. It has advantages like being applicable to both homogeneous and inhomogeneous problems and simplifying calculations.
This document discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
First order non-linear partial differential equation & its applicationsJayanshu Gundaniya
There are five types of methods for solving first order non-linear partial differential equations:
I) Equations containing only p and q variables. II) Equations relating z as a function of u. III) Equations that can be separated into functions of single variables. IV) Clairaut's Form where the solution is directly substituted. V) Charpit's Method which is a general method taking integrals of auxiliary equations to solve dz=pdx+qdy and find the solution. These types cover a range of applications including Poisson's, Helmholtz's, and Schrödinger's equations in fields like electrostatics, elasticity, wave theory and quantum mechanics.
Series solution of ordinary differential equation
advance engineering mathematics
The power series method is the standard method for solving linear ODEs with variable
coefficients. It gives solutions in the form of power series. These series can be used for computing values, graphing curves, proving formulas, and exploring properties of
solutions, as we shall see.
The document discusses concepts related to partial differentiation and its applications. It covers topics like tangent planes, linear approximations, differentials, Taylor expansions, maxima and minima problems, and the Lagrange method. Specifically, it defines the tangent plane to a surface at a point using partial derivatives, describes how to find the linear approximation of functions, and explains how to find maximum and minimum values of functions using critical points and the second derivative test.
This document contains a summary of a student group project on first order ordinary differential equations (ODEs). It defines key terms related to ODEs such as order, degree, general solutions, and singular solutions. It also categorizes common types of first order ODEs including separable, homogeneous, exact, and linear equations. Solution methods are described for each type. Additional topics covered include Bernoulli equations, orthogonal trajectories, and applications of ODEs in areas like radioactivity, electrical circuits, economics, and physics. The document is authored by six chemical engineering students at G.H. Patel College of Engineering and Technology.
Changing variable is something we come across very often in Integration. There are many
reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
Eigen values and eigen vectors engineeringshubham211
mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function
...
This document discusses higher order differential equations and their applications. It introduces second order homogeneous differential equations and their solutions based on the nature of the roots. Non-homogeneous differential equations are also discussed, along with their general solution being the sum of the solution to the homogeneous equation and a particular solution. Methods for solving non-homogeneous equations are presented, including undetermined coefficients and reduction of order. Applications to problems in various domains like physics, engineering, and circuits are also outlined.
Polar coordinates represent points in a plane using an ordered pair (r, θ) where r is the distance from a fixed point called the pole (or origin) and θ is the angle between the line from the pole to the point and a reference axis called the polar axis. Some key properties:
- Points (r, θ) and (-r, θ + π) represent the same location.
- Polar equations like r = a describe circles with radius a centered at the pole.
- Equations can be converted between Cartesian (x, y) and polar (r, θ) coordinates using trigonometric relationships.
- Graphs of polar equations r = f(θ) consist of all points satisfying
The document discusses the gamma function, which generalizes the factorial function to real and complex numbers. It provides a brief history of the gamma function, noting it was first introduced by Euler to extend the factorial to non-integer values. The document defines the gamma function, provides some of its key properties like its relationship to the factorial of integers, and discusses other related functions like the beta function. It also gives some examples of applications of the gamma function, such as using it to calculate volumes of n-dimensional balls and in computing infinite sums.
The document defines a homogeneous linear differential equation as an equation of the form:
a0(dx/dy)n + a1(dx/dy)n-1 + ... + an-1(dx/dy) + any = X, where a0, a1, ..., an are constants and X is a function of x.
It provides the method of solving such equations by first reducing it to a linear equation with constant coefficients, then taking a trial solution of the form y = emz, and finally solving the resulting auxiliary equation.
It proves identities relating derivatives of y with respect to x and z, then uses these identities to solve two sample homogeneous linear differential equations of orders 2 and
This document discusses Bessel's equation and its solutions. It begins by defining Bessel's equation of order ν. It then examines specific cases for ν = 0, 1/2, and 1. For each case it derives the recurrence relation and finds the first solution, which is a Bessel function of the first kind for that order. It also discusses the Bessel function of the second kind for order zero. The document provides graphs and discusses approximations of Bessel functions for large values of x.
Power Series - Legendre Polynomial - Bessel's EquationArijitDhali
The presentation shows types of equations inside every topic along with its general form, generating formula, and other equations like recursion, frobenius, rodrigues etc for calculus. Its an overall explanation in a brief. You are at correct link to get your work done out of this in your engineering maths.
This document provides an introduction to ordinary differential equations (ODEs). It defines ODEs as differential equations containing functions of one independent variable and its derivatives. The document discusses some key concepts related to ODEs including order, degree, and different types of ODEs such as variable separable, homogeneous, exact, linear, and Bernoulli's equations. Examples of each type of ODE are provided along with the general methods for solving each type.
Second order homogeneous linear differential equations Viraj Patel
1) The document discusses second order linear homogeneous differential equations, which have the general form P(x)y'' + Q(x)y' + R(x)y = 0.
2) It describes methods for finding the general solution including reduction of order, and discusses the solutions when the coefficients are constants.
3) The general solution depends on the nature of the roots of the auxiliary equation: distinct real roots, repeated real roots, or complex roots.
Partial differential equation & its application.isratzerin6
Partial differential equations (PDEs) involve partial derivatives of dependent variables with respect to more than one independent variable. PDEs can be linear if the dependent variable and all its partial derivatives occur linearly, or non-linear. PDEs are used to model systems in fields like physics, engineering, and quantum mechanics, with examples being the Laplace, heat, and wave equations used in fluid dynamics, heat transfer, and quantum mechanics respectively. The heat equation specifically describes the distribution of heat over time in a given region.
The document discusses Fourier series and two of their applications. Fourier series can be used to represent periodic functions as an infinite series of sines and cosines. This allows approximating functions that are not smooth using trigonometric polynomials. Two key applications are representing forced oscillations, where a periodic driving force can be modeled as a Fourier series, and solving the heat equation, where the method of separation of variables results in a Fourier series representation of temperature over space and time.
The document discusses complex differentiability and analytic functions. It shows that for a function f(z) to be complex differentiable, the Cauchy-Riemann equations relating the partial derivatives of the real and imaginary parts must be satisfied. It also discusses representing functions as power series and their radii of convergence. Multivalued functions like logarithms and roots are discussed, noting the need for branch cuts to define single-valued branches.
The document discusses Bessel functions, which are solutions to a second order differential equation that arises in diverse situations. It also discusses the Hankel transform, which expresses functions as a weighted sum of Bessel functions of the first kind. The Hankel transform is useful for problems with cylindrical or spherical symmetry, as it appears when taking the multidimensional Fourier transform in hyperspherical coordinates. It has advantages like being applicable to both homogeneous and inhomogeneous problems and simplifying calculations.
This document discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
First order non-linear partial differential equation & its applicationsJayanshu Gundaniya
There are five types of methods for solving first order non-linear partial differential equations:
I) Equations containing only p and q variables. II) Equations relating z as a function of u. III) Equations that can be separated into functions of single variables. IV) Clairaut's Form where the solution is directly substituted. V) Charpit's Method which is a general method taking integrals of auxiliary equations to solve dz=pdx+qdy and find the solution. These types cover a range of applications including Poisson's, Helmholtz's, and Schrödinger's equations in fields like electrostatics, elasticity, wave theory and quantum mechanics.
Series solution of ordinary differential equation
advance engineering mathematics
The power series method is the standard method for solving linear ODEs with variable
coefficients. It gives solutions in the form of power series. These series can be used for computing values, graphing curves, proving formulas, and exploring properties of
solutions, as we shall see.
The document discusses concepts related to partial differentiation and its applications. It covers topics like tangent planes, linear approximations, differentials, Taylor expansions, maxima and minima problems, and the Lagrange method. Specifically, it defines the tangent plane to a surface at a point using partial derivatives, describes how to find the linear approximation of functions, and explains how to find maximum and minimum values of functions using critical points and the second derivative test.
This document contains a summary of a student group project on first order ordinary differential equations (ODEs). It defines key terms related to ODEs such as order, degree, general solutions, and singular solutions. It also categorizes common types of first order ODEs including separable, homogeneous, exact, and linear equations. Solution methods are described for each type. Additional topics covered include Bernoulli equations, orthogonal trajectories, and applications of ODEs in areas like radioactivity, electrical circuits, economics, and physics. The document is authored by six chemical engineering students at G.H. Patel College of Engineering and Technology.
Changing variable is something we come across very often in Integration. There are many
reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
Eigen values and eigen vectors engineeringshubham211
mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function
...
This document discusses higher order differential equations and their applications. It introduces second order homogeneous differential equations and their solutions based on the nature of the roots. Non-homogeneous differential equations are also discussed, along with their general solution being the sum of the solution to the homogeneous equation and a particular solution. Methods for solving non-homogeneous equations are presented, including undetermined coefficients and reduction of order. Applications to problems in various domains like physics, engineering, and circuits are also outlined.
Polar coordinates represent points in a plane using an ordered pair (r, θ) where r is the distance from a fixed point called the pole (or origin) and θ is the angle between the line from the pole to the point and a reference axis called the polar axis. Some key properties:
- Points (r, θ) and (-r, θ + π) represent the same location.
- Polar equations like r = a describe circles with radius a centered at the pole.
- Equations can be converted between Cartesian (x, y) and polar (r, θ) coordinates using trigonometric relationships.
- Graphs of polar equations r = f(θ) consist of all points satisfying
The document discusses the gamma function, which generalizes the factorial function to real and complex numbers. It provides a brief history of the gamma function, noting it was first introduced by Euler to extend the factorial to non-integer values. The document defines the gamma function, provides some of its key properties like its relationship to the factorial of integers, and discusses other related functions like the beta function. It also gives some examples of applications of the gamma function, such as using it to calculate volumes of n-dimensional balls and in computing infinite sums.
The document defines a homogeneous linear differential equation as an equation of the form:
a0(dx/dy)n + a1(dx/dy)n-1 + ... + an-1(dx/dy) + any = X, where a0, a1, ..., an are constants and X is a function of x.
It provides the method of solving such equations by first reducing it to a linear equation with constant coefficients, then taking a trial solution of the form y = emz, and finally solving the resulting auxiliary equation.
It proves identities relating derivatives of y with respect to x and z, then uses these identities to solve two sample homogeneous linear differential equations of orders 2 and
This document discusses Bessel's equation and its solutions. It begins by defining Bessel's equation of order ν. It then examines specific cases for ν = 0, 1/2, and 1. For each case it derives the recurrence relation and finds the first solution, which is a Bessel function of the first kind for that order. It also discusses the Bessel function of the second kind for order zero. The document provides graphs and discusses approximations of Bessel functions for large values of x.
Power Series - Legendre Polynomial - Bessel's EquationArijitDhali
The presentation shows types of equations inside every topic along with its general form, generating formula, and other equations like recursion, frobenius, rodrigues etc for calculus. Its an overall explanation in a brief. You are at correct link to get your work done out of this in your engineering maths.
The document discusses series solutions to second order linear differential equations near ordinary points. It provides an example of finding the series solution to the differential equation y'' + y = 0 near x0 = 0. The solution is found to be a cosine series which represents the cosine function, a fundamental solution. A second example finds the series solution to Airy's equation near x0 = 0, obtaining fundamental solutions related to Airy functions.
This document discusses series solutions and special functions in engineering mathematics. It covers:
1) Finding series solutions to second order differential equations with variable coefficients by expressing the solution as a power series.
2) Bessel equations and their series solutions, including the general solutions for non-integer and integer orders of the Bessel equation.
3) Properties of Bessel functions, Legendre polynomials, and their generating functions. This includes orthogonality relations and recurrence relations.
Differential equation and Laplace transformsujathavvv
- Differential equations relate an unknown function and its derivatives, and are classified as ordinary (ODE) or partial (PDE) depending on the number of independent variables. Higher order equations can be reduced to systems of first order equations.
- Exact differential equations can be written as the total differential of a function, while non-exact equations may require an integrating factor to convert them to exact form.
- The general solution to a non-homogeneous second order differential equation is the sum of the complementary function (solution to the homogeneous equation) and a particular integral similar to the non-homogeneous term.
Differential equation and Laplace transformMohanamalar8
- Differential equations relate an unknown function and its derivatives, and are classified as ordinary (ODE) or partial (PDE) depending on the number of independent variables. Higher order equations can be reduced to systems of first order equations.
- Exact differential equations can be written as the total differential of a function, while non-exact equations may require an integrating factor to convert them to exact form.
- The general solution to a non-homogeneous second order differential equation is the sum of the complementary function (solution to the homogeneous equation) and a particular integral similar to the non-homogeneous term.
Digital Text Book :POTENTIAL THEORY AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS Baasilroy
This document summarizes potential theory and elliptic partial differential equations. It discusses Laplace's equation, which is a second order partial differential equation that is elliptic in nature. The document covers solutions to Laplace's equation, including the general solution for two and three variables. It also discusses Green's identities and theorems related to harmonic functions, including the maximum principle, minimum principle, and uniqueness theorem.
The document summarizes an analytic approach to solving the Collatz conjecture proposed by Berg and Meinardus. It introduces two linear operators, U and V, acting on the space of holomorphic functions on the open unit disk. The kernel K of U and V, defined as the set of functions where both operators equal 0, is of main interest. If it can be shown that K equals the two-dimensional space Δ2, then the Collatz conjecture would be proven true. The individual kernel KV of V is already known from prior work. The paper aims to compute U[h] for h in KV and show that K = Δ2, which would imply the truth of the Collatz conjecture.
This document provides an introduction to associated Legendre functions and spherical harmonics. It discusses how the spherical harmonics arise in solving partial differential equations with spherical symmetry, such as the Laplace, heat, wave, and Schrodinger equations. The key properties of associated Legendre functions and spherical harmonics are summarized, including their definitions, differential equations, orthogonality relations, and recurrence relations. Several examples of low-order spherical harmonics are also provided.
The document discusses continuity of functions and operations on continuous functions. It begins by defining continuity of a function at a point and gives examples of determining continuity. It then discusses right and left continuity. Graphical interpretations of continuity are provided. The concept of extending a function by continuity is introduced along with examples. Finally, the document defines continuity over an interval and states properties of operations on continuous functions.
This document summarizes methods for solving ordinary differential equations (ODEs). It discusses:
1) Types of ODEs including order, degree, linear/nonlinear.
2) Four methods for solving 1st order ODEs: separable variables, homogeneous equations, exact equations, and integrating factors.
3) Solutions to higher order linear ODEs using complementary functions and particular integrals.
4) Finding complementary functions and particular integrals for ODEs with constant coefficients.
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.
The document discusses several topics related to calculus including:
1. How to graph functions using GeoGebra by inputting the function and seeing the outputted graph and table of values.
2. How to find the derivative of composite functions using formulas for derivatives of sums, differences, products, and quotients of functions.
3. The definitions of even and odd functions and how to determine if a given function is even or odd based on its behavior when the input is negated.
4. How to evaluate limit problems by directly substituting the limit value or using trigonometric limits such as lim x->0 sin(x)/x = 1.
The document presents a multi-step problem to determine the area bounded by a circle, the x-axis, y-axis, and the line y=12. It provides the equation of the circle and guides the reader through simplifying systems of equations to find values for variables in the circle equation. These are used to graph the circle and identify the bounded area as a rectangle minus one quarter of the circle area, calculated as 66.9027 square units.
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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 topic on matrix theory and linear algebra is fundamental. The focus is on subjects like systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices that will be helpful in other fields.
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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.
A Proof of Hypothesis Riemann and it is proven that apply only for equation ζ(z)=0.Also here it turns out that it does not apply in General Case.(DOI:10.13140/RG.2.2.10888.34563/5)
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...EduSkills OECD
Andreas Schleicher, Director of Education and Skills at the OECD presents at the launch of PISA 2022 Volume III - Creative Minds, Creative Schools on 18 June 2024.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
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2. Bessel Functions
The differential equation
where p is a non negative constant, is called Bessels
equation, and its solutions are known as Bessel
functions.
These functions first arose in Daniel Bernoulli’s
investigation of the oscillations of a hanging chain
appeared again in Euler’s theory of the vibration of a
circular membrane and Besell’s studies of planetary
motion.
,0)( 222
=−+′+′′ ypxyxyx
3. Bessel Functions continued…
Friedrich Wilhelm Bessel (1784 – 1846)
studied disturbances in planetary motion,
which led him in 1824 to make the first
systematic analysis of solutions of this
equation.
The solutions became known as Bessel
functions.
4. Bessel Functions continued…
Bessel’s Equation of order p:
The origin that is x = 0 is a regular singular point.
The indicial equation is m2
-p2
=0, and the exponents are
m1=p and m2 = -p.
It follows from Theorem 30-A that BE has a solution of
the form
where and the power series converges
for all x.
( ) .0222
=−+′+′′ ypxyxyx
∑∑
∞
=
+
∞
=
==
00
,
n
pn
n
n
n
n
p
xaxaxy
00 ≠a ∑ n
n xa
5. Bessel Functions continued…
Hence we have
The terms of the equation becomes
( )
( )( )∑
∑ ∑
∞
=
−+
∞
=
∞
=
−++
−++=′′
+=′=
0
2
0 0
1
.1)(
,)(,)(
n
pn
n
n n
pn
n
pn
n
xpnpnaxy
xpnaxyxaxy
( ) ( )( )∑∑
∑ ∑
∞
=
+
∞
=
+
∞
=
∞
=
+
−
+
−++=′′+=′
=−=−
0
2
0
0 0
2
222
.1,
,,
n
pn
n
n
pn
n
n n
pn
n
pn
n
xpnpnayxxpnayx
xayxxapyp
6. Bessel Functions continued…
Inserting into the equation and equating the coefficients
of xn+p
to zero, we get the following recursion formula
We know that a0 is arbitrary and a-1=0 tells us that a1=0;
and repeated application of this recursion formula yields
the fact that an=0 for every odd subscript n.
.
n)pn(
a
a
aanpn
n
n
nn
+
−=
=++
−
−
2
or
0)2(
2
2
7. Bessel Functions continued…
The nonzero coefficients of our solution are therefore
And the solution is
,
)62)(42)(22(642)62(6
,
)42)(22(42)42(4
,
)22(2
,
04
6
02
4
0
20
+++⋅⋅
−=
+
−=
++⋅
=
+
−=
+
−=
ppp
a
p
a
a
pp
a
p
a
a
p
a
aa
+
+++
−
++
+
+
−
=
)3)(2)(1(!32
)2)(1(!22)1(2
1
6
6
4
4
2
2
0
ppp
x
pp
x
p
x
xay p
8. Bessel Functions continued…
Or we have
This solution y is known as Bessel function of the first
kind of order p.
This is denoted by Jp(x) and is defined by replacing the
arbitrary constant a0 by . So we have
.
)()2)(1(!2
)1(
0
2
2
0 ∑
∞
= +++
−=
n
n
n
np
npppn
x
xay
)!2/(1 pp
∑
∑
∞
=
+
∞
=
+
−=
+++
−=
0
2
0
2
2
.
)!(!
)2/(
)1(
)()2)(1(!2
)1(
!2
)(
n
pn
n
n
n
n
n
p
p
p
npn
x
npppn
x
p
x
xJ
9. Bessel Functions continued…
The most useful Bessel functions are those of order 0
and 1 which are
And
+
⋅⋅
−
⋅
+−=
−= ∑
∞
=
222
6
22
4
2
2
0
2
0
642422
1
!!
)2/(
)1()(
xxx
nn
x
xJ
n
n
n
−
+
−=
+
−=
+∞
=
∑
53
12
0
1
2!3!2
1
2!2!1
1
2
2)!1(!
1
)1()(
xxx
x
nn
xJ
n
n
n
11. Bessel Functions continued…
These graphs displays several interesting properties of
the functions J0(x) and J1(x); each has a damped
oscillating behavior producing an infinite number of
positive zeros; and these zeros occur alternatively, in a
manner suggesting the functions cos x and sin x.
(positive zeros- the positive real numbers for which the
function Jp(x) vanishes).
We have seen that in the denominator of Jp(x) there is a
term (p+n)! , but it is meaning less if p is not a positive
integer.
Now our next turn is to overcome this difficulty.
12. Bessel Functions continued…
The Gamma Function:
The gamma function is defined by
The factor so rapidly as that this
integral converges at the upper limit regardless of the
value of p. However at the lower limit we have
and the factor whenever p<1.
The restriction that p must be positive is necessary in
order to guarantee convergence at the lower limit.
∫
∞
−−
>=Γ
0
1
.0,)( pdtetp tp
0→−t
e ∞→t
1→−t
e
∞→−1p
t
13. Bessel Functions continued…
We can see that
For integration by parts yields
Since as .
);()1( ppp Γ=+Γ
).(lim
lim
lim)1(
0
1
0
1
0
0
ppdtetp
dtetpet
dtetp
b
tp
b
b
tpbtp
b
b
tp
b
Γ=
=
+−=
=+Γ
∫
∫
∫
−−
∞→
−−−
∞→
−
∞→
0/ →bp
eb ∞→b
14. Bessel Functions continued…
If we use the fact that
Then the formula yields
And in general for any integer greater
than equal to zero.
Its extension to other values of p.
Now with the help of the formula
,1)1(
0
==Γ ∫
∞
−
dte t
);()1( ppp Γ=+Γ
,!3123)3(3)4(,!212)2(2)3(,1)1(1)2( =⋅⋅=Γ=Γ=⋅=Γ=Γ=Γ=Γ
!)1( nn =+Γ
.
)1(
)(
p
p
p
+Γ
=Γ
15. Bessel Functions continued…
We extend it to other values of p (it is necessary for the
applications).
If –1<p<0, then 0<p+1<1, so the right side of the
equation G1 has a value and the left side of G1 is
defined to have the value given by right side.
The next step is that when –2<p<-1, then –1<p+1<0, so
we can use the formula G1 to define on the
interval –2<p<-1 in terms of the values of .
We can continue this process indefinitely.
.
)1(
)(
p
p
p
+Γ
=Γ G1
)( pΓ
)1( +Γ p
16. Bessel Functions continued…
Also we can see that
Accordingly as from the right or left. This
function behaves in a similar way near all negative
integers. Its graph is shown in the next slide.
Gamma function gives us
Since never vanishes the function will
be defined and well behaved for all values of p, if we
agree that .
±∞=
+Γ
=Γ
→→ p
p
p
pp
)1(
lim)(lim
00
0→p
)( pΓ
.
2
1
π=
Γ
)( pΓ )(/1 pΓ
,2,1,0for0)(/1 −−==Γ pp
17. Bessel Functions continued…
These ideas enable us to define p! by
for all values of p except negative integers.
Also
is defined for all values of p and has the value 0 when
ever p is negative integer.
Now the Bessel function Jp(x) defined by the formula
has a meaning for all values of p greater than equal to 0.
)1(! +Γ= pp
)1(
1
!
1
+Γ
=
pp
∑
∞
=
+
+
−=
0
2
.
)!(!
)2/(
)1()(
n
pn
n
p
npn
x
xJ
19. Bessel Functions continued…
The general Solution of Bessel’s Equation:
We have found one particular solution for the exponent
m1=p, namely Jp(x).
To get the general solution we must have to construct a
second linearly independent solution, that is one which
is not a constant multiple of Jp(x).
Any such solution is called a Bessel function of
second kind.
For the second LI solution the procedure is to try the
other exponent m2=-p, but in doing this we expect to
encounter difficulties whenever the difference m1-m2
=2p is zero or a positive integer.
20. Bessel Functions continued…
The difficulties are serious if the difference is zero. So
we assume p is not an integer. We replace p by –p in our
earlier treatment.
We see the change in
and if it happens that p=1/2, then by letting n=1 we see
that there is no compulsion to choose a1=0.
But since we want a particular solution it is permissible
to put a1=0. The same problem arise when p=3/2 and
n=3, and so on; and we solve it by putting a1=a3=… =0.
;0)2( 2 =++− −nn aanpn
21. Bessel Functions continued…
Other calculations goes as before. Hence we obtain the
second solution as
the first term of the series is
so J-p(x) is unbounded near x=0. Since Jp(x) is bounded
near x=0, these two solutions are independent and
is the general solution of Bessel’s equation.
,
)!(!
)2/(
)1()(
2
0 npn
x
xJ
pn
n
n
p
+−
−=
−∞
=
− ∑
,
2)!(
1
p
x
p
−
−
integer,annot),()( 21 pxJcxJcy pp −+=
B2
B3
22. Bessel Functions continued…
The solution is entirely different when p is an integer
Formula B2 becomes
since the factors 1/(-m+n)! are zero when n=0,1,…, m-1.
On replacing the dummy variable n by n+m and
compensating by beginning the summation at n=0.
.0≠m
)!(!
)2/(
)1(
)!(!
)2/(
)1()(
2
2
0
nmn
x
nmn
x
xJ
mn
mn
n
mn
n
n
m
+−
−=
+−
−=
−∞
=
−∞
=
−
∑
∑
23. Bessel Functions continued…
We obtain
This shows that J-m(x) is not independent of Jm(x) so in
this case
is not the general solution.
).()1(
)!(!
)2/(
)1()1(
!)!(
)2/(
)1()(
2
0
)(2
0
xJ
nmn
x
nmn
x
xJ
m
m
mn
n
nm
mmn
n
mn
m
−=
+
−−=
+
−=
+∞
=
−+∞
=
+
−
∑
∑
),()( 21 xJcxJcy mm −+=
24. Bessel Functions continued…
One possible approach is by Section 16, which yield
As a second solution independent of Jm(x).
When p is not an integer any function of the form B3
with is a Bessel function of the second kind
including J-p(x) it self.
The standard Bessel function of second kind is defined
by
∫ 2
)(
)(
xxJ
dx
xJ
m
m
02 ≠c
.
sin
)(cos)(
)(
π
π
p
xJpxJ
xY
pp
p
−−
=
25. Bessel Functions continued…
This choice of Yp is made for two reasons:
(We have a problem when p is an integer but we avoid it
by taking the limit which is explained in the first reason)
First reason: (for p is an integer we take the limit)
We can see that (detail analysis is omitted) the function
exists and is Bessel functions of the second kind and it
follows that
is the general solution of Bessel's equation in all cases,
whether p is an integer or not.
)(lim)( xYxY p
mp
m
→
=
)()( 21 xYcxJcy pp +=
26. Bessel Functions continued…
Second reason:
By introducing a new dependent variable
we transform the Bessel equation into
When x is very large, the above equation closely
approximates the familiar differential equation
Which has independent solutions
We therefore expect that for large values of x, any
Bessel function y(x) will behave like some linear
combination of
),()( xyxxu =
.0
4
41
1 2
2
=
−
++′′ u
x
p
u
.0=+′′ uu
.sin)(andcos)( 21 xxuxxu ==
27. Bessel Functions continued…
And this expectation is supported by the fact
and
where r1(x) and r2(x) are bounded as
.sin
1
cos
1
x
x
andx
x
2/3
1 )(
24
cos
2
)(
x
xrp
x
x
xJ p +
−−=
ππ
π
,
)(
24
sin
2
)( 2/3
2
x
xrp
x
x
xYp +
−−=
ππ
π
.∞→x
29. Bessel Functions continued…
Properties of Bessel Functions:
The Bessel function Jp(x) is defined for any real number
p by
Identities and the function Jm+1/2(x).
We prove the following identity
∑
∞
=
+
+
−=
0
2
.
)!(!
)2/(
)1()(
n
pn
n
p
npn
x
xJ
[ ] )()( 1 xJxxJx
dx
d
p
p
p
p
−= I1
30. Bessel Functions continued…
Proof:
[ ]
)(
)!1(!
)2/()1(
)!1(!2
)1(
)!(!2
)1(
)(
1
0
12
0
12
122
0
2
22
xJx
npn
x
x
npn
x
npn
x
dx
d
xJx
dx
d
p
p
n
pnn
p
n
pn
pnn
n
pn
pnn
p
p
−
∞
=
−+
∞
=
−+
−+
∞
=
+
+
=
−+
−
=
−+
−
=
+
−
=
∑
∑
∑
31. Bessel Functions continued…
And the second identity is
Proof:
[ ] ).()( 1 xJxxJx
dx
d
p
p
p
p
+
−−
−= I2
[ ]
).(
)!1(!
)2/()1(
)!()!1(2
)1(
)!(!2
)1(
)(
1
0
12
1
12
12
0
2
2
xJx
npn
x
x
npn
x
npn
x
dx
d
xJx
dx
d
p
p
n
pnn
p
n
pn
nn
n
pn
nn
p
p
+
−
∞
=
++
−
∞
=
−+
−
∞
=
+
−
−=
++
−
−=
+−
−
=
+
−
=
∑
∑
∑
32. Bessel Functions continued…
If the differentiation is carried out and the results are
divided by , then the formula becomes
If I3 and I4 are added we get
p
x±
)()()(
and
)()()(
1
1
xJxJ
x
p
xJ
xJxJ
x
p
xJ
ppp
ppp
+
−
−=−′
=+′ I3
I4
),()()(2 11 xJxJxJ ppp +− −=′ I5
33. Bessel Functions continued…
and if subtracted, we get
or
These formulas enables us to write Bessel functions and
their derivatives in terms of other Bessel functions.
An interesting applications begins with identity 6 and
with the formulas (already established in the previous
exercise)
).()()(
2
11 xJxJxJ
x
p
ppp +− += I6
,cos
2
)(sin
2
)( 2/12/1 x
x
xJandx
x
xJ
ππ
== −
).(
2
)()( 11 xJ
x
p
xJxJ ppp +−= −+
34. Bessel Functions continued…
From I6 it follows that
and
also
and
−=−= − x
x
x
x
xJxJ
x
xJ cos
sin2
)()(
1
)( 2/12/12/3
π
.sin
cos3sin32
)()(
3
)( 22/12/32/5
−−=−= x
x
x
x
x
x
xJxJ
x
xJ
π
.sin
cos2
)()(
1
)( 2/12/12/3
−−=−−= −− x
x
x
x
xJxJ
x
xJ
π
.cos
cos3cos32
)()(
3
)( 22/12/32/5
−+=−−= −−− x
x
x
x
x
x
xJxJ
x
xJ
π
35. Bessel Functions continued…
It can be continued indefinitely, and therefore every
Bessel function Jm+1/2(x) (where m is an integer) is
elementary.
It has been proved by Liouville that these functions are
the only cases in which Jp(x) is elementary.
The I6 yields Lamberts continued fraction
for tan x, and this continued fraction led to the
first proof of the fact that is not a rational
number.
π
36. Bessel Functions continued…
When the differentiation formulas I1 and I2 are written
in the form
Then they serve for the integration of many simple
expressions containing Bessel functions. For example,
when p=1, we get
∫
∫
+−=
+=
−
+
−
−
,)()(
and
,)()(
1
1
cxJxdxxJx
cxJxdxxJx
p
p
p
p
p
p
p
p
∫ += .)()( 10 cxxJdxxxJ
37. Bessel Functions continued…
Zeros and Bessel Series:
For every value of p, the function Jp(x) has an infinite
number of positive zeros.
For J0(x) the first five positive zeros are
2.4048, 5.5201, 8.6537, 11.7915, and 14.9309.
For J1(x) the first five positive zeros are
3.8317, 7.0156, 10.1735, 13.3237, and 16.4706.
The purpose of concern of these zeros of Jp(x) is:
It is often necessary in mathematical physics to expand
a given function in terms of Bessel functions.
38. Bessel Functions continued…
The simplest and most useful expansion of this kind are
the series of the form
Where f(x) is defined on the interval and the
are the positive zeros of some fixed Bessel function
Jp(x) with p greater than equal to zero.
As in the previous experience in Legendre series here
also we need to determine the coefficients of the
expansion, which depend on certain integral properties
of the function .
+++== ∑
∞
=
)()()()()( 3322
1
11 xJaxJaxJaxJaxf pp
n
pnpn λλλλ
10 ≤≤ x nλ
)( xJ np λ
39. Bessel Functions continued…
Here we need the fact
The function are said to be orthogonal with
respect to the weight function x on the interval
If the expansion is assumed to be possible then
multiplying through , and integrating term by
term from 0 to 1, and using the above fact we get
∫
=
≠
=
+
1
0
2
1 .)(
2
1
,0
)()(
nmifJ
nmif
dxxJxxJ
np
npmp
λ
λλ
)( xJ np λ
.10 ≤≤ x
)( xxJ mp λ
∫ +=
1
0
2
1 ;)(
2
)()( mp
m
mp J
a
dxxJxxf λλ
40. Bessel Functions continued…
and on replacing m by n we obtain the formula for the
coefficients an as:
The series
With the coefficients given by the above formula is
know as Bessel series (or sometimes the Fourier-Bessel
series) of the function f(x).
∫+
=
1
02
1
.)()(
)(
2
dxxJxxf
J
a np
np
n λ
λ
+++== ∑
∞
=
)()()()()( 3322
1
11 xJaxJaxJaxJaxf pp
n
pnpn λλλλ
41. Bessel Functions continued…
This theorem is given without proof which tells about
the conditions under which the series actually converges.
Theorem A. (Bessel expansion theorem).
Assume that f(x) and have at most finite
number of jump discontinuities on the interval
. If 0<x<1, then the Bessel series
converges to f(x) when x is a point of continuity
of this function, and converges to
when x is a point of discontinuity.
)(xf ′
10 ≤≤ x
[ ])()(
2
1
++− xfxf
42. Bessel Functions continued…
At the end point x=1 the series converges to zero
regardless of the nature of the function because every
is zero. The series also converges at x=0, to zero if p>1
and to f(0+) if p=0.
Example 1.
Bessel series of the function in terms
of the functions .
Solution: We use this formula for the coefficients
)( npJ λ
,10,1)( ≤≤= xxf
)(0 xJ nλ
∫+
=
1
02
1
.)()(
)(
2
dxxJxxf
J
a np
np
n λ
λ
43. Bessel Functions continued…
But
By using the formula
so we get
Now it follows that
is the desired Bessel series.
,
)(
)(
1
)()()(
1
1
0
1
1
0
1
0
00
n
n
n
n
nn
J
xxJ
dxxxJdxxJxxf
λ
λ
λ
λ
λλ
=
=
=∫ ∫
∑
∞
=
≤≤==
1
0
1
)10()(
)(
2
1)(
n
n
nn
xxJ
J
xf λ
λλ
∫ +=− ,)()(1 cxJxdxxJx p
p
p
p
)(
2)(
)(
2
)()(
)(
2
1
1
2
1
1
02
1
nnn
n
n
np
np
n
J
J
J
dxxJxxf
J
a
λλλ
λ
λ
λ
λ
==
= ∫+
44. Bessel Functions continued…
Proofs of Orthogonality Properties
To establish
We begin with the fact that y=Jp(x) is a solution of
If a and b are two distinct positive constants, it follows
that the functions u(x)=Jp(ax) and v(x)=Jp(bx) satisfy the
equations
∫
=
≠
=
+
1
0
2
1 .)(
2
1
,0
)()(
nmifJ
nmif
dxxJxxJ
np
npmp
λ
λλ
.01
1
2
2
=
−+′+′′ y
x
p
y
x
y
45. Bessel Functions continued…
and
We now multiply these equations by v and u and
subtract the results, to obtain
and after multiplying by x, this becomes
0
1
2
2
2
=
−+′+′′ u
x
p
au
x
u
.0
1
2
2
2
=
−+′+′′ v
x
p
bv
x
v
;)()(
1
)( 22
uvabuvvu
x
uvvu
dx
d
−=′−′+′−′
.)()]([ 22
xuvabuvvux
dx
d
−=′−′
46. Bessel Functions continued…
When this expression is integrated from x=0 to x=1, we get
The expression in brackets vanishes at x=0 and at the other
end of the interval we have u(1)=Jp(a) and v(1) = Jp(b).
So when x=1 the right side becomes
It follows that the integral on the left is zero if a and b are
distinct positive zeros and of Jp(x); that is we have
which is the first part of the proof.
.)]([)( 1
0
1
0
22
uvvuxdxxuvab ′−′=− ∫
∫ =
1
0
,0)()( dxxJxxJ npmp λλ
mλ nλ
).()()()( aubvbvau ′−′
47. Bessel Functions continued…
Now we have to evaluate the integral when m=n. If
is multiplied by , it becomes
or
so on integrating form x=0 to x=1,
0
1
2
2
2
=
−+′+′′ u
x
p
au
x
u
ux ′2
2
02222 22222
=′−′+′+′′′ uupuuxauxuux
,0)(2)()( 222222222
=−−+′ up
dx
d
xuauxa
dx
d
ux
dx
d
48. Bessel Functions continued…
we obtain
when x=0. The expression in brackets vanishes; and
since we get
Now putting we get
since Jp(a)=0 and
and the proof of the second part is complete.
,])([2 1
0
222222
1
0
22
upxauxdxxua −+′=∫
),()1( aJau p
′=′
.)(1
2
1
)(
2
1
)( 2
2
2
2
1
0
2
aJ
a
p
aJdxaxxJ ppp
−+′=∫
na λ=
,)(
2
1
)(
2
1
)( 2
1
2
1
0
2
npnpnp JJdxxxJ λλλ +=′=∫
)()()()()( 11 npnpppp JJxJxJ
x
p
xJ λλ ++ =
′
⇒=−
′
49. Bessel Functions continued…
Exercise 1: Show that
And between any two zeros of J0(x) there is a zero of
J1(x), and also between any two zeros of J1(x) there is a
zero of J0(x).
Solution: Taking the derivative of J0(x) we get
).()]([and);()( 0110 xxJxxJ
dx
d
xJxJ
dx
d
=−=
[ ]
).(
)!1(!
)2/()1(
)!1(!
)2/()1(
!!2
2)1(
!!2
)1(
)(
1
0
121
1
12
1
2
12
0
2
2
0
xJ
nn
x
nn
x
nn
nx
nn
x
dx
d
xJ
dx
d
n
nn
n
nn
n
n
nn
n
n
nn
−=
+
−
=
−
−
=
−
=
−
=
∑∑
∑∑
∞
=
++∞
=
−
∞
=
−∞
=
50. Bessel Functions continued…
Suppose x1 and x2 are any two positive zeros of J0(x), then
J0(x1)=0=J0(x2) and J0(x) is differentiable for all positive
values of x.
But by Rolle’s theorem if f(x) is continuous on [a,b] and
differentiable on (a,b) and f(a)=f(b)=0 then there exists at
least one number c in (a,b) such that
Hence applying this theorem there exists at least one
number x3 in (x1, x2) such that
Hence in between any two positive zeros of J0(x) there is
a zero for J1(x). Other part is left to you.
.0)( =′ cf
.0)(.0)()(or0)( 31313030 ==−=
′
=
′
xJeixJxJxJ
51. Bessel Functions continued…
Exercise 2: Express J2(x), J3(x) in terms of J0(x) and
J1(x).
Solution:Using the formula
We get
Similarly others.
).(
2
)()( 11 xJ
x
p
xJxJ ppp +−= −+
).()(
2
)(
12
)()()(
01
10211
xJxJ
x
xJ
x
xJxJxJ
−=
⋅
+−==+
52. Bessel Functions continued…
Exercise 3: If f(x)=xp
, for the interval show that
its Bessel series in the functions where the
are the positive zeros of Jp(x) is
Solution: The coefficients are given by the formula
Hence replacing f(x) by xp
and calculating the
coefficients we get
10 ≤≤ x
)( xJp nλ nλ
∑
∞
= +
=
1 1
).(
)(
2
n
np
npn
p
xJ
J
x λ
λλ
∫+
=
1
02
1
.)()(
)(
2
dxxJxxf
J
a np
np
n λ
λ
53. Bessel Functions continued…
The coefficient an as
Putting the coefficient in the series we get
Which completes the proof.
.
)(
2)(
)(
2
)(
)(
2
)(
)(
2
1
1
2
1
1
01
1
2
1
1
0
1
2
1
npnn
np
np
np
n
p
np
np
p
np
n
J
J
J
xJ
x
J
dxxJx
J
a
λλλ
λ
λ
λ
λλ
λ
λ
+
+
+
+
+
+
+
+
==
=
= ∫
∑∑
∞
= +
∞
=
===
1 11
)(
)(
2
)()(
n
np
npnn
npn
p
xJ
J
xJaxxf λ
λλ
λ
54. Bessel Functions continued…
Exercise 4: Prove that
Solution:
[ ] [ ].)()()()(
2
1
2
1 xJxJxxJxxJ
dx
d
pppp ++ −=
[ ] [ ]
[ ] [ ]
[ ]
[ ].)()(
)()()()(
)()()()(
)()()()(
2
1
2
11
11
1
1
1
1
1
1
1
xJxJx
xJxxJxxJxxJx
xJx
dx
d
xJxxJx
dx
d
xJx
xJxxJx
dx
d
xJxxJ
dx
d
pp
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
pp
+
+
−
+
++−
−
+
+
+
+−
+
+−
+
−=
−+=
+=
=