This document discusses functions of a complex variable. It introduces complex numbers and their representations. It covers topics like complex differentiation using Cauchy-Riemann equations, analytic functions, Cauchy's integral theorem, and contour integrals. Functions of a complex variable provide tools for physics concepts involving complex quantities like wavefunctions. Cauchy's integral theorem states that the contour integral of an analytic function over a closed path is zero.
- A differential equation involves 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 exponent of the highest order derivative.
- Linear differential equations involve the dependent variable and its derivatives only to the first power. Non-linear equations do not meet this criterion.
- The general solution of a differential equation contains as many arbitrary constants as the order of the equation. A particular solution results from assigning values to the arbitrary constants.
- Differential equations can be solved through methods like variable separation, inspection of reducible forms, and finding homogeneous or linear representations.
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 line integrals in the complex plane. It defines line integrals, shows how complex line integrals are equivalent to two real line integrals, and reviews how to parameterize curves to evaluate line integrals. It also covers Cauchy's theorem, which states that the line integral of an analytic function around a closed curve in its domain is zero. The fundamental theorem of calculus for complex variables and the Cauchy integral formula are also summarized.
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.
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
This document discusses the topic of differentiation. It begins by defining differentiation and listing some fundamental rules, such as the chain rule and differentiation of constants. It then discusses geometrically what the derivative represents at a point and lists several types of differentiable functions. The document goes on to explain differentiation using substitution, of implicit functions, and of parametric functions. It also covers successive differentiation, Leibnitz's theorem, and differentiation of special function types. The document provides an overview of differentiation concepts and rules.
This document provides an overview of triple integrals and their applications. It defines triple integrals as the limit of triple Riemann sums for functions of three variables, analogous to double integrals. Triple integrals can be evaluated over rectangular boxes by expressing them as iterated integrals in any of six orders, as stated by Fubini's theorem. The document also describes how to evaluate triple integrals over more general bounded solid regions, including type 1 regions bounded by two graphs, type 2 regions bounded between two planes, and type 3 regions. It provides examples of evaluating triple integrals over a tetrahedron and other specific regions.
Verification of Solenoidal & IrrotationalMdAlAmin187
This document discusses vector analysis and defines solenoidal and irrotational vector functions. It provides examples to verify whether specific vector functions are solenoidal or irrotational. Specifically:
It defines key concepts in vector analysis including vectors, vector operations, and vector-valued functions. It then discusses the history and development of vector analysis.
It defines a solenoidal (divergence-free) vector function as one where the divergence is equal to zero. An example verifies that a given vector function is not solenoidal.
It defines an irrotational (curl-free) vector function as one where the curl is equal to zero. An example verifies that a given vector function is irrotational
- A differential equation involves 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 exponent of the highest order derivative.
- Linear differential equations involve the dependent variable and its derivatives only to the first power. Non-linear equations do not meet this criterion.
- The general solution of a differential equation contains as many arbitrary constants as the order of the equation. A particular solution results from assigning values to the arbitrary constants.
- Differential equations can be solved through methods like variable separation, inspection of reducible forms, and finding homogeneous or linear representations.
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 line integrals in the complex plane. It defines line integrals, shows how complex line integrals are equivalent to two real line integrals, and reviews how to parameterize curves to evaluate line integrals. It also covers Cauchy's theorem, which states that the line integral of an analytic function around a closed curve in its domain is zero. The fundamental theorem of calculus for complex variables and the Cauchy integral formula are also summarized.
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.
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
This document discusses the topic of differentiation. It begins by defining differentiation and listing some fundamental rules, such as the chain rule and differentiation of constants. It then discusses geometrically what the derivative represents at a point and lists several types of differentiable functions. The document goes on to explain differentiation using substitution, of implicit functions, and of parametric functions. It also covers successive differentiation, Leibnitz's theorem, and differentiation of special function types. The document provides an overview of differentiation concepts and rules.
This document provides an overview of triple integrals and their applications. It defines triple integrals as the limit of triple Riemann sums for functions of three variables, analogous to double integrals. Triple integrals can be evaluated over rectangular boxes by expressing them as iterated integrals in any of six orders, as stated by Fubini's theorem. The document also describes how to evaluate triple integrals over more general bounded solid regions, including type 1 regions bounded by two graphs, type 2 regions bounded between two planes, and type 3 regions. It provides examples of evaluating triple integrals over a tetrahedron and other specific regions.
Verification of Solenoidal & IrrotationalMdAlAmin187
This document discusses vector analysis and defines solenoidal and irrotational vector functions. It provides examples to verify whether specific vector functions are solenoidal or irrotational. Specifically:
It defines key concepts in vector analysis including vectors, vector operations, and vector-valued functions. It then discusses the history and development of vector analysis.
It defines a solenoidal (divergence-free) vector function as one where the divergence is equal to zero. An example verifies that a given vector function is not solenoidal.
It defines an irrotational (curl-free) vector function as one where the curl is equal to zero. An example verifies that a given vector function is irrotational
Cauchy integral theorem & formula (complex variable & numerical method )Digvijaysinh Gohil
1) The document discusses the Cauchy Integral Theorem and Formula. It states that if a function f(z) is analytic inside and on a closed curve C, then the integral of f(z) around C is equal to 0.
2) It provides examples of evaluating integrals using the Cauchy Integral Theorem when the singularities lie outside the closed curve C.
3) The Cauchy Integral Formula is introduced, which expresses the value of an analytic function F(a) inside C as a contour integral around C. Examples are worked out applying this formula to find the value and derivatives of functions at points inside C.
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.
The document presents information about differential equations including:
- A definition of a differential equation as an equation containing the derivative of one or more variables.
- Classification of differential equations by type (ordinary vs. partial), order, and linearity.
- Methods for solving different types of differential equations such as variable separable form, homogeneous equations, exact equations, and linear equations.
- An example problem demonstrating how to use the cooling rate formula to calculate the time of death based on measured body temperatures.
This document discusses approaches to teaching complex numbers. It describes an axiomatic approach, utilitarian approach, and historical approach. The historical approach builds on prior knowledge of quadratic equations and introduces complex numbers to solve problems like finding the roots of quadratic and cubic equations. The document also covers definitions of complex numbers, addition, subtraction, multiplication, and division of complex numbers. It discusses pedagogical considerations like using multiple representations and building on students' prior knowledge.
Line integral,Strokes and Green TheoremHassan Ahmed
The document defines key concepts related to line integrals of vector fields. It defines a vector as having both magnitude and direction. It then defines a line integral as integrating a function along a line, and defines a vector field as a region where a vector quantity (like magnetic field) assigns a unique vector value to each point. Finally, it discusses the definition of a line integral of a vector field, and three fundamental theorems relating line integrals to other integrals: the gradient theorem relating it to differences of a potential function at endpoints, Green's theorem relating it to a double integral, and Stokes' theorem relating it to a surface integral.
The document discusses vector calculus concepts including:
1) Coordinate systems used in vector calculus problems including rectangular, cylindrical, and spherical coordinates.
2) How to write vectors and their components in each coordinate system.
3) Relationships between vectors in different coordinate systems using transformation matrices.
4) Concepts of gradient, divergence, and curl and their definitions and representations in different coordinate systems.
5) Theorems relating integrals, including the divergence theorem and Stokes' theorem.
This document provides an overview of Laplace transforms. Key points include:
- Laplace transforms convert differential equations from the time domain to the algebraic s-domain, making them easier to solve. The process involves taking the Laplace transform of each term in the differential equation.
- Common Laplace transforms of functions are presented. Properties such as linearity, differentiation, integration, and convolution are also covered.
- Partial fraction expansion is used to break complex fractions in the s-domain into simpler forms with individual terms that can be inverted using tables of transforms.
- Solving differential equations using Laplace transforms follows a standard process of taking the Laplace transform of each term, rewriting the equation in the s-domain, solving
The double integral of a function f(x,y) over a bounded region R in the xy-plane is defined as the limit of Riemann sums that approximate the total value of f over R. This double integral is denoted by the integral of f(x,y) over R and its value is independent of the subdivision used in the Riemann sums. Properties and methods for evaluating double integrals are discussed, along with applications such as finding the area, volume, mass, and moments of inertia. Changes of variables in double integrals using the Jacobian are also covered.
The document discusses various topics related to complex functions and complex analysis. It defines concepts such as distance between complex numbers, circles, circular disks, neighborhoods, annuli, open and closed sets, connected sets, domains, regions, bounded regions, single-valued and multi-valued functions, and limits and continuity of complex functions. Specific examples are provided to illustrate definitions of circles, neighborhoods, single-valued and multi-valued functions. The limit of a complex function as z approaches a point z0 is defined using the epsilon-delta definition of a limit.
This document discusses different types of vector integrals: line integrals, surface integrals, and volume integrals. It provides details on line integrals, including that they integrate a function along a curve rather than a straight line, and can integrate either scalar or vector fields. Line integrals of scalar fields give the area under the curve, while line integrals of vector fields geometrically represent the total displacement along the curve. Surface and volume integrals are also introduced but not described in detail.
This document provides an overview of complex analysis, including:
1) Limits and their uniqueness in complex analysis, such as the limit of a function f(z) as z approaches z0.
2) The definition of a continuous function in complex analysis as one where the limit exists at each point in the domain and equals the function value.
3) Analytic functions, which are differentiable in some neighborhood of each point in their domain.
1. The document discusses vector calculus concepts including the gradient, divergence, curl, and theorems relating integrals.
2. It defines the curl of a vector field A as the maximum circulation of A per unit area and provides expressions for curl in Cartesian, cylindrical and spherical coordinates.
3. Stokes's theorem is described as relating a line integral around a closed path to a surface integral of the curl over the enclosed surface, allowing transformation between different integral types.
1) Stokes' theorem relates a surface integral over a surface S to a line integral around the boundary curve of S. It states that the line integral of a vector field F around a closed curve C that forms the boundary of a surface S is equal to the surface integral of the curl of F over the surface S.
2) In Example 1, Stokes' theorem is used to evaluate a line integral around an elliptical curve C by calculating the corresponding surface integral over the elliptical region S bounded by C.
3) In Example 2, Stokes' theorem is again used, this time to evaluate a line integral around a circular curve C by calculating the surface integral over the part of a sphere bounded by C.
The document discusses the Gauss Divergence Theorem, which states that the volume integral of the divergence of a vector field over a volume is equal to the surface integral of that vector field over the bounding surface of the volume. The divergence of a vector field at a point represents the flux of that vector field diverging out per unit volume at that point. The divergence can be positive, negative, or zero, indicating whether there are sources, sinks, or neither of the vector field at that point.
This document discusses complex variables and functions. It covers topics such as:
- Cauchy-Riemann conditions, which must be satisfied for a complex function to be analytic/differentiable
- Cauchy's integral theorem, which states that the integral of an analytic function around a closed contour is zero
- Harmonic functions, whose real and imaginary parts satisfy the 2D Laplace equation
- Examples of calculating contour integrals of functions like zn along circular and square paths
The document provides proofs of theorems like Cauchy's integral theorem using techniques like Stokes' theorem. It also discusses simply and multiply connected regions in relation to contour integrals.
Gram schmidt orthogonalization | Orthonormal Process Isaac Yowetu
The document discusses Gram-Schmidt orthogonalization and orthonormalization. It defines orthogonalization as constructing orthogonal vectors that span a subspace, while orthonormalization results in unit vectors. The Gram-Schmidt process is described as a method to take a set of vectors and construct an orthogonal set from them. Two examples applying the Gram-Schmidt process are shown.
Analytic Function, C-R equation, Harmonic function, laplace equation, Construction of analytic function, Critical point, Invariant point , Bilinear Transformation
In this lecture, we will discuss:
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
The document is an introduction to ordinary differential equations prepared by Ahmed Haider Ahmed. It defines key terms like differential equation, ordinary differential equation, partial differential equation, order, degree, and particular and general solutions. It then provides methods for solving various types of first order differential equations, including separable, homogeneous, exact, linear, and Bernoulli equations. Specific examples are given to illustrate each method.
Complex Variable Interest Entity [sanitized] Michael Burgess
- Water Valley, LLC was formed by Crocker Wellspring, LLC, ABC Tax Credit Equity, LLC, and ABC Tax Credit Manager II, Inc. to develop an affordable housing apartment complex in Atlanta, Georgia.
- Crocker Wellspring contributed $100 and was named Manager. ABC Tax Credit Manager II contributed $10 and was named Special Member. ABC Tax Credit Equity contributed $3,739,463 and was named Investor Member.
- ABC Tax Credit Equity's capital contributions of $3,739,463 would be paid out in seven installments upon the Company meeting certain requirements at each stage of development. The capital contributions were subject to potential downward adjustments based on the certified tax credits and delays
ELECTRÓNICA+RADIO+TV. Tomo II.: VÁLVULAS DE VACÍO I. ELECTROMETRÍA TEÓRICO-PR...Gabriel Araceli
ELECTRÓNICA+RADIO+TV. Tomo II: VÁLVULAS DE VACÍO I. ELECTROMETRÍA TEÓRICO-PRÁCTICA.
Lección 10: Aparatos para medir corrientes continuas. Galvanómetros de cuadro móvil.
Lección 11: Medición de c.a. Aparatos de medida para c.a. Tipos de c.a. Lecturas correctas. Construcción de un tester (segunda fase)
Lección 12: Medida de resistencias. Ohmetros. Medición de capacidades. El capacímetro. Medida de autoinducciones. Construcción de un tester (final).
Esta obra perteneció a un curso a distancia durante los años 60-70 y se encuentra descatalogada.La tecnología empleada, por tanto, ha quedado obsoleta, pero la teoría permanece y está expuesta con una pedagogía excelente. Es una obra básica para los estudiantes y digna de figurar en la biblioteca de cualquier profesional de la electrónica. Por ello me he tomado el trabajo de escanearlos y ponerlos a disposición de aquellos a los que pueda interesar. Febrero de 2017.
Cauchy integral theorem & formula (complex variable & numerical method )Digvijaysinh Gohil
1) The document discusses the Cauchy Integral Theorem and Formula. It states that if a function f(z) is analytic inside and on a closed curve C, then the integral of f(z) around C is equal to 0.
2) It provides examples of evaluating integrals using the Cauchy Integral Theorem when the singularities lie outside the closed curve C.
3) The Cauchy Integral Formula is introduced, which expresses the value of an analytic function F(a) inside C as a contour integral around C. Examples are worked out applying this formula to find the value and derivatives of functions at points inside C.
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.
The document presents information about differential equations including:
- A definition of a differential equation as an equation containing the derivative of one or more variables.
- Classification of differential equations by type (ordinary vs. partial), order, and linearity.
- Methods for solving different types of differential equations such as variable separable form, homogeneous equations, exact equations, and linear equations.
- An example problem demonstrating how to use the cooling rate formula to calculate the time of death based on measured body temperatures.
This document discusses approaches to teaching complex numbers. It describes an axiomatic approach, utilitarian approach, and historical approach. The historical approach builds on prior knowledge of quadratic equations and introduces complex numbers to solve problems like finding the roots of quadratic and cubic equations. The document also covers definitions of complex numbers, addition, subtraction, multiplication, and division of complex numbers. It discusses pedagogical considerations like using multiple representations and building on students' prior knowledge.
Line integral,Strokes and Green TheoremHassan Ahmed
The document defines key concepts related to line integrals of vector fields. It defines a vector as having both magnitude and direction. It then defines a line integral as integrating a function along a line, and defines a vector field as a region where a vector quantity (like magnetic field) assigns a unique vector value to each point. Finally, it discusses the definition of a line integral of a vector field, and three fundamental theorems relating line integrals to other integrals: the gradient theorem relating it to differences of a potential function at endpoints, Green's theorem relating it to a double integral, and Stokes' theorem relating it to a surface integral.
The document discusses vector calculus concepts including:
1) Coordinate systems used in vector calculus problems including rectangular, cylindrical, and spherical coordinates.
2) How to write vectors and their components in each coordinate system.
3) Relationships between vectors in different coordinate systems using transformation matrices.
4) Concepts of gradient, divergence, and curl and their definitions and representations in different coordinate systems.
5) Theorems relating integrals, including the divergence theorem and Stokes' theorem.
This document provides an overview of Laplace transforms. Key points include:
- Laplace transforms convert differential equations from the time domain to the algebraic s-domain, making them easier to solve. The process involves taking the Laplace transform of each term in the differential equation.
- Common Laplace transforms of functions are presented. Properties such as linearity, differentiation, integration, and convolution are also covered.
- Partial fraction expansion is used to break complex fractions in the s-domain into simpler forms with individual terms that can be inverted using tables of transforms.
- Solving differential equations using Laplace transforms follows a standard process of taking the Laplace transform of each term, rewriting the equation in the s-domain, solving
The double integral of a function f(x,y) over a bounded region R in the xy-plane is defined as the limit of Riemann sums that approximate the total value of f over R. This double integral is denoted by the integral of f(x,y) over R and its value is independent of the subdivision used in the Riemann sums. Properties and methods for evaluating double integrals are discussed, along with applications such as finding the area, volume, mass, and moments of inertia. Changes of variables in double integrals using the Jacobian are also covered.
The document discusses various topics related to complex functions and complex analysis. It defines concepts such as distance between complex numbers, circles, circular disks, neighborhoods, annuli, open and closed sets, connected sets, domains, regions, bounded regions, single-valued and multi-valued functions, and limits and continuity of complex functions. Specific examples are provided to illustrate definitions of circles, neighborhoods, single-valued and multi-valued functions. The limit of a complex function as z approaches a point z0 is defined using the epsilon-delta definition of a limit.
This document discusses different types of vector integrals: line integrals, surface integrals, and volume integrals. It provides details on line integrals, including that they integrate a function along a curve rather than a straight line, and can integrate either scalar or vector fields. Line integrals of scalar fields give the area under the curve, while line integrals of vector fields geometrically represent the total displacement along the curve. Surface and volume integrals are also introduced but not described in detail.
This document provides an overview of complex analysis, including:
1) Limits and their uniqueness in complex analysis, such as the limit of a function f(z) as z approaches z0.
2) The definition of a continuous function in complex analysis as one where the limit exists at each point in the domain and equals the function value.
3) Analytic functions, which are differentiable in some neighborhood of each point in their domain.
1. The document discusses vector calculus concepts including the gradient, divergence, curl, and theorems relating integrals.
2. It defines the curl of a vector field A as the maximum circulation of A per unit area and provides expressions for curl in Cartesian, cylindrical and spherical coordinates.
3. Stokes's theorem is described as relating a line integral around a closed path to a surface integral of the curl over the enclosed surface, allowing transformation between different integral types.
1) Stokes' theorem relates a surface integral over a surface S to a line integral around the boundary curve of S. It states that the line integral of a vector field F around a closed curve C that forms the boundary of a surface S is equal to the surface integral of the curl of F over the surface S.
2) In Example 1, Stokes' theorem is used to evaluate a line integral around an elliptical curve C by calculating the corresponding surface integral over the elliptical region S bounded by C.
3) In Example 2, Stokes' theorem is again used, this time to evaluate a line integral around a circular curve C by calculating the surface integral over the part of a sphere bounded by C.
The document discusses the Gauss Divergence Theorem, which states that the volume integral of the divergence of a vector field over a volume is equal to the surface integral of that vector field over the bounding surface of the volume. The divergence of a vector field at a point represents the flux of that vector field diverging out per unit volume at that point. The divergence can be positive, negative, or zero, indicating whether there are sources, sinks, or neither of the vector field at that point.
This document discusses complex variables and functions. It covers topics such as:
- Cauchy-Riemann conditions, which must be satisfied for a complex function to be analytic/differentiable
- Cauchy's integral theorem, which states that the integral of an analytic function around a closed contour is zero
- Harmonic functions, whose real and imaginary parts satisfy the 2D Laplace equation
- Examples of calculating contour integrals of functions like zn along circular and square paths
The document provides proofs of theorems like Cauchy's integral theorem using techniques like Stokes' theorem. It also discusses simply and multiply connected regions in relation to contour integrals.
Gram schmidt orthogonalization | Orthonormal Process Isaac Yowetu
The document discusses Gram-Schmidt orthogonalization and orthonormalization. It defines orthogonalization as constructing orthogonal vectors that span a subspace, while orthonormalization results in unit vectors. The Gram-Schmidt process is described as a method to take a set of vectors and construct an orthogonal set from them. Two examples applying the Gram-Schmidt process are shown.
Analytic Function, C-R equation, Harmonic function, laplace equation, Construction of analytic function, Critical point, Invariant point , Bilinear Transformation
In this lecture, we will discuss:
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
The document is an introduction to ordinary differential equations prepared by Ahmed Haider Ahmed. It defines key terms like differential equation, ordinary differential equation, partial differential equation, order, degree, and particular and general solutions. It then provides methods for solving various types of first order differential equations, including separable, homogeneous, exact, linear, and Bernoulli equations. Specific examples are given to illustrate each method.
Complex Variable Interest Entity [sanitized] Michael Burgess
- Water Valley, LLC was formed by Crocker Wellspring, LLC, ABC Tax Credit Equity, LLC, and ABC Tax Credit Manager II, Inc. to develop an affordable housing apartment complex in Atlanta, Georgia.
- Crocker Wellspring contributed $100 and was named Manager. ABC Tax Credit Manager II contributed $10 and was named Special Member. ABC Tax Credit Equity contributed $3,739,463 and was named Investor Member.
- ABC Tax Credit Equity's capital contributions of $3,739,463 would be paid out in seven installments upon the Company meeting certain requirements at each stage of development. The capital contributions were subject to potential downward adjustments based on the certified tax credits and delays
ELECTRÓNICA+RADIO+TV. Tomo II.: VÁLVULAS DE VACÍO I. ELECTROMETRÍA TEÓRICO-PR...Gabriel Araceli
ELECTRÓNICA+RADIO+TV. Tomo II: VÁLVULAS DE VACÍO I. ELECTROMETRÍA TEÓRICO-PRÁCTICA.
Lección 10: Aparatos para medir corrientes continuas. Galvanómetros de cuadro móvil.
Lección 11: Medición de c.a. Aparatos de medida para c.a. Tipos de c.a. Lecturas correctas. Construcción de un tester (segunda fase)
Lección 12: Medida de resistencias. Ohmetros. Medición de capacidades. El capacímetro. Medida de autoinducciones. Construcción de un tester (final).
Esta obra perteneció a un curso a distancia durante los años 60-70 y se encuentra descatalogada.La tecnología empleada, por tanto, ha quedado obsoleta, pero la teoría permanece y está expuesta con una pedagogía excelente. Es una obra básica para los estudiantes y digna de figurar en la biblioteca de cualquier profesional de la electrónica. Por ello me he tomado el trabajo de escanearlos y ponerlos a disposición de aquellos a los que pueda interesar. Febrero de 2017.
ELECTRÓNICA+RADIO+TV. Tomo IV: AMPLIFICADORES B.F. ALTAVOCES. VÁLVULAS AMPLIFICADORAS.
Lección 23: Distorsión de amplitud y de frecuencia. Capacidades parásitas. Curva de respuesta de un amplificador. Teorema de Fourier. Amplificador de B.F. tipo comercial, calidad Hi-Fi.
Lección 24: Los controles de tono. Grabación y reproducción de discos. Estudio de un amplificador para tocadiscos. Estudio práctico de una maleta tocadiscos.
Lección 25: Montajes del triodo. El seguidor catódico. Amplificadores de c.c. Amplificadores en contrafase. Amplificadores clase A, AB, B y C.
Esta obra perteneció a un curso a distancia durante los años 60-70 y se encuentra descatalogada.La tecnología empleada, por tanto, ha quedado obsoleta, pero la teoría permanece y está expuesta con una pedagogía excelente. Es una obra básica para los estudiantes y digna de figurar en la biblioteca de cualquier profesional de la electrónica. Febrero de 2017.
Guad2D is a two-dimensional hydraulic simulation model designed to analyse freshet waves caused by rain or the gradual or spontaneous destruction of dams and flood walls in large water deposits.
This document discusses the key functions of management. It describes management as a social, continuous, universal, and interrelated process. The main functions of management are identified as planning, organizing, staffing, directing, coordinating, and controlling. Planning involves setting objectives and developing strategies to achieve goals. Organizing requires grouping and assigning work, and delegating authority and responsibility. Staffing deals with acquiring and maintaining qualified human resources. Directing functions include leadership, communication, motivation, and supervision of employees. Coordinating arranges collective efforts to achieve unity of action. Controlling monitors performance and ensures plans are followed, making corrections if needed.
1) The Reserve Bank of India (RBI) is India's central bank. It was established in 1935 and nationalized in 1949. RBI performs traditional central banking functions like issuing currency, acting as a banker to the government and banks, and maintaining foreign exchange reserves.
2) RBI also regulates the banking system through tools like the cash reserve ratio (CRR), which requires banks to hold a portion of deposits with RBI, the repo rate at which banks borrow from RBI, and the reverse repo rate at which RBI borrows from banks.
3) Other RBI functions include acting as a lender of last resort, controlling credit in the economy, collecting and publishing banking data, and
Complex problem solving in nederland, een studieCoThink
Het World Economic Forum ziet complex problem solving als de belangrijkste vaardigheid voor werknemers, nu en in de toekomst. Hoe zit dat in Nederland? Michel den Otter en David Veldhoen deden een onderzoek...
The document discusses potential titles for a new music magazine. The working title is "Overload," which is meant to convey that the magazine will be packed with images, gossip and information. While "Overload" grabs attention, it does not clearly indicate the music genres covered. The author plans to choose a one-word masthead in line with popular music magazine conventions, and will further develop the title as the magazine concept progresses.
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1.[1] In this expression, a is the real part and b is the imaginary part of the complex number. If {\displaystyle z=a+bi} {\displaystyle z=a+bi}, then {\displaystyle \Re z=a,\quad \Im z=b.} {\displaystyle \Re z=a,\quad \Im z=b.}
Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers are a field extension of the ordinary real numbers, in order to solve problems that cannot be solved with real numbers alone.
There are typically three levels of management in organizations: top-level managers who set goals for the organization; middle-level managers who carry out top-level goals and set goals for departments; and first-level managers who manage employees on a daily basis. Managers at different levels perform different amounts of planning, organizing, leading, and controlling. They also fill different decisional, interpersonal, and informational roles. Many organizations are flattening hierarchies to allow for faster decision-making and better communication. This results in fewer middle management positions.
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iTutor provides information on complex numbers. Complex numbers consist of real and imaginary parts and can be written as a + bi, where a is the real part and b is the imaginary part. The imaginary unit i = √-1. Properties of complex numbers include: the square of i is -1; complex conjugates are obtained by changing the sign of the imaginary part; and the basic arithmetic operations of addition, subtraction, and multiplication follow predictable rules when applied to complex numbers. Complex numbers allow representing solutions, like the square root of a negative number, that are not possible with real numbers alone.
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This document discusses complex numbers and functions. It introduces complex numbers using Cartesian (x + iy) and polar (r(cosθ + i sinθ)) forms. It describes the Cauchy-Riemann conditions that must be satisfied for a function of a complex variable to be differentiable. A function is analytic if it satisfies the Cauchy-Riemann conditions and its partial derivatives are continuous. Analytic functions have properties like equality of second-order partial derivatives and establishing a relation between the real and imaginary parts.
1) Complex numbers can be represented in Cartesian (x + iy) or polar (r(cosθ + i sinθ)) form, with conversions between the two.
2) The derivative of a complex function f(z) is defined if the Cauchy-Riemann equations are satisfied.
3) A function is analytic if it is differentiable and its partial derivatives are continuous, implying the Cauchy-Riemann equations always hold. Analytic functions have properties like equality of second partial derivatives.
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1. The document introduces complex numbers and some basic results regarding complex numbers such as the complex conjugate and modulus of a complex number.
2. It then discusses functions of a complex variable, defining a complex function and its Cartesian and polar forms. It also covers continuity, derivatives, and analytic functions of a complex variable.
3. The Cauchy-Riemann equations are derived and provide a necessary condition for a function to be analytic (differentiable everywhere in a neighborhood). Two examples are provided to illustrate the Cauchy-Riemann equations and analytic functions.
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This document discusses functions of complex variables. It defines key concepts such as complex numbers, the Cauchy-Riemann conditions for differentiability, and analytic functions. The Cauchy-Riemann conditions require the partial derivatives of the real and imaginary parts of a complex function to satisfy certain relationships. Functions that satisfy the Cauchy-Riemann conditions everywhere are said to be analytic. The document also discusses singularities such as poles and essential singularities.
This document discusses complex functions and their derivatives. It defines a complex function as a function f(z) that maps complex numbers to complex numbers. The derivative of a complex function is defined as the limit of the difference quotient, which may not always exist. Some simple functions like Re(z) are shown to not have complex derivatives. The usual rules of differentiation, such as the sum, product, quotient and chain rules, are shown to hold for complex differentiable functions.
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Complex Analysis And ita real life problems solutionNaeemAhmad289736
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SWOT analysis in the project Keeping the Memory @live.pptx
Unit1
1. Functions of a Complex Variable
UNIT-1
Dr. M K Singh
Associate Professor
Jahangirabad Institute of Technology,
Barabanki
2. Functions of A Complex Variables I
Functions of a complex variable provide us some powerful and
widely useful tools in theoretical physics.
• Some important physical quantities are complex variables (the
wave-function )
• Evaluating definite integrals.
• Obtaining asymptotic solutions of differentials
equations.
• Integral transforms
• Many Physical quantities that were originally real become complex
as simple theory is made more general. The energy
( the finite life time).
iEE nn
0
/1
3. We here go through the complex algebra briefly.
A complex number z = (x,y) = x + iy, Where.
We will see that the ordering of two real numbers (x,y) is significant,
i.e. in general x + iy y + ix
X: the real part, labeled by Re(z); y: the imaginary part, labeled by Im(z)
Three frequently used representations:
(1) Cartesian representation: x+iy
(2) polar representation, we may write
z=r(cos + i sin) or
r – the modulus or magnitude of z
- the argument or phase of z
1i
i
erz
4. r – the modulus or magnitude of z
- the argument or phase of z
The relation between Cartesian
and polar representation:
The choice of polar representation or Cartesian representation is a
matter of convenience. Addition and subtraction of complex variables
are easier in the Cartesian representation. Multiplication, division,
powers, roots are easier to handle in polar form,
1/ 22 2
1
tan /
r z x y
y x
21
2121
i
errzz
21
2121 //
i
errzz
innn
erz
z1 ± z2 = (x1 ± x2 )+i(y1 ± y2 )
z1z2 = (x1x2 - y1y2 )+i(x1y2 + x2y1)
5. From z, complex functions f(z) may be constructed. They can be
written
f(z) = u(x,y) + iv(x,y)
in which v and u are real functions.
For example if , we have
The relationship between z and f(z) is best pictured as a
mapping operation, we address it in detail later.
)arg()arg()arg( 2121 zzzz
2121 zzzz
xyiyxzf 222
Using the polar form,
2
)( zzf
6. Function: Mapping operation
x
y Z-plane
u
v
The function w(x,y)=u(x,y)+iv(x,y) maps points in the xy plane into points
in the uv plane.
nin
i
ie
ie
)sin(cos
sincos
We get a not so obvious formula
Since
n
inin )sin(cossincos
7. Complex Conjugation: replacing i by –i, which is denoted by (*),
We then have
Hence
Note:
ln z is a multi-valued function. To avoid ambiguity, we usually set n=0
and limit the phase to an interval of length of 2. The value of lnz with
n=0 is called the principal value of lnz.
iyxz *
222*
ryxzz
21*
zzz Special features: single-valued function of a
real variable ---- multi-valued function
i
rez ni
re 2
irlnzln nirz 2lnln
9. Analytic functions
If f(z) is differentiable at and in some small region around ,
we say that f(z) is analytic at
Differentiable: Cauthy-Riemann conditions are satisfied
the partial derivatives of u and v are continuous
Analytic function:
Property 1:
Property 2: established a relation between u and v
022
vu
Example:
Find the analytic functions w(z) = u(x, y)+iv(x, y)
if (a) u(x, y) = x3
-3xy2
;(v = 3x3
y- y3
+c)
(b) v(x, y) = e-y
sin x;(v = ?)
0zz
0zz
0z
10. Cauchy-Riemann Equations
0 0 0
0 0
0
0
0
1 0 0
2 0 0
Let , , be diff. at
then lim exists
with
In particular, can be computed along
: , i.e.
: , i.e.
z
f z u x y iv x y z x iy
f z z f z
f z
z
z x i y
f z
C y y x x z x
C x x y y
z i y
11. Cauchy-Riemann Equations
0 0 0 0
0
0 0 0 0
( , ) ( , )
( , ) ( , )
u v
x y i x y
x x
f z
u v
i x y x y
y y
13. Theorem
A necessary condition for a fun.
f(z)=u(x,y)+iv(x,y)
to be diff. at a point z0 is that the C-R eq. hold at
z0.
Consequently, if f is analytic in an open set G,
then the C-R eq. must hold at every point of G.
14. Theorem
A necessary condition for a fun.
f(z)=u(x,y)+iv(x,y)
to be diff. at a point z0 is that the C-R eq. hold at
z0.
Consequently, if f is analytic in an open set G,
then the C-R eq. must hold at every point of G.
15. Application of Theorem
To show that a function is NOT analytic, it
suffices to show that the C-R eq. are not
satisfied
16. Cauchy – Riemann conditions
Having established complex functions, we now proceed to
differentiate them. The derivative of f(z), like that of a real function, is
defined by
provided that the limit is independent of the particular approach to the
point z. For real variable, we require that
Now, with z (or zo) some point in a plane, our requirement that the
limit be independent of the direction of approach is very restrictive.
Consider
zf
dz
df
z
zf
z
zfzzf
zz
00
limlim
o
xxxx
xfxfxf
oo
limlim
yixz
viuf
,
yix
viu
z
f
17. Let us take limit by the two different approaches as in the figure. First,
with y = 0, we let x0,
Assuming the partial derivatives exist. For a second approach, we set
x = 0 and then let y 0. This leads to
If we have a derivative, the above two results must be identical. So,
x
v
i
x
u
z
f
xz
00
limlim
x
v
i
x
u
y
v
y
u
i
z
f
z
0
lim
y
v
x
u
,
x
v
y
u
18. These are the famous Cauchy-Riemann conditions. These Cauchy-
Riemann conditions are necessary for the existence of a derivative, that
is, if exists, the C-R conditions must hold.
Conversely, if the C-R conditions are satisfied and the partial
derivatives of u(x,y) and v(x,y) are continuous, exists.
xf
zf
19. Cauchy’s integral Theorem
We now turn to integration.
in close analogy to the integral of a real function
The contour is divided into n intervals .Let
with for j. Then
'
00 zz
01 jjj zzz
0
0
1
lim
z
z
n
j
jj
n
dzzfzf
n
The right-hand side of the above equation is called the contour (path) integral
of f(z)
.and
bewteencurveon thepointaiswhere
,andpointsthechoosing
ofdetailstheoftindependen
isandexistslimitthat theprovided
1
j
j
jj
j
zz
z
20. As an alternative, the contour may be defined by
with the path C specified. This reduces the complex integral to the
complex sum of real integrals. It’s somewhat analogous to the case of
the vector integral.
An important example
22
11
2
1
,,
yx
yxc
z
zc
idydxyxivyxudzzf
22
11
22
11
yx
yx
yx
yxcc
udyvdxivdyudx
c
n
dzz
where C is a circle of radius r>0 around the origin z=0 in the
direction of counterclockwise.
21. In polar coordinates, we parameterize
and , and have
which is independent of r.
Cauchy’s integral theorem
– If a function f(z) is analytical (therefore single-valued) [and its partial
derivatives are continuous] through some simply connected region R, for
every closed path C in R,
i
rez
diredz i
2
0
1
1exp
22
1
dni
r
dzz
i
n
c
n
1-nfor1
-1nfor0
{
0 dzzf
c
22. •Multiply connected regions
The original statement of our theorem demanded a simply connected
region. This restriction may easily be relaxed by the creation of a
barrier, a contour line. Consider the multiply connected region of
Fig.1.6 In which f(z) is not defined for the interior R
Cauchy’s integral theorem is not valid for the contour C, but we can
construct a C for which the theorem holds. If line segments DE and
GA arbitrarily close together, then
E
D
A
G
dzzfdzzf
24. Cauchy’s Integral Formula
Cauchy’s integral formula: If f(z) is analytic on and within a closed contour C
then
in which z0 is some point in the interior region bounded by C. Note that
here z-z0 0 and the integral is well defined.
Although f(z) is assumed analytic, the integrand (f(z)/z-z0) is not
analytic at z=z0 unless f(z0)=0. If the contour is deformed as in Fig.1.8
Cauchy’s integral theorem applies.
So we have
0
0
2 zif
zz
dzzf
C
C C
dz
zz
zf
zz
dzzf
2
0
00
25. Let , here r is small and will eventually be made to
approach zero
(r0)
Here is a remarkable result. The value of an analytic function is given at
an interior point at z=z0 once the values on the boundary C are
specified.
What happens if z0 is exterior to C?
In this case the entire integral is analytic on and within C, so the
integral vanishes.
i
0 rezz
drie
re
rezf
dz
zz
dzzf i
C C
i
i
2 2
0
0
00 2
2
zifdzif
C
26. Derivatives
Cauchy’s integral formula may be used to obtain an expression for
the derivation of f(z)
Moreover, for the n-th order of derivative
0
0 0
1
2
f z dzd
f z
dz i z z
Ñ
C
zf
zz
dzzf
i exteriorz,0
interiorz,
2
1
0
00
0
2
000 2
11
2
1
zz
dzzf
izzdz
d
dzzf
i
1
0
0
2
!
n
n
zz
dzzf
i
n
zf
28. In the above case, on a circle of radius r about the origin,
then (Cauchy’s inequality)
Proof:
where
Lowville's theorem: If f(z) is analytic and bounded in the complex
plane, it is a constant.
Proof: For any z0, construct a circle of radius R around z0,
Mzf
Mra n
n
nn
rz
nn
r
M
r
r
rM
z
dzzf
a
11
2
2
2
1
rfMaxrM rz
22
0
0
2
22
1
R
RM
zz
dzzf
i
zf
R
R
M
29. Since R is arbitrary, let , we have
Conversely, the slightest deviation of an analytic function from a
constant value implies that there must be at least one singularity
somewhere in the infinite complex plane. Apart from the trivial constant
functions, then, singularities are a fact of life, and we must learn to live
with them, and to use them further.
R
.const)z(f,e.i,0zf
30. Laurent Expansion
Taylor Expansion
Suppose we are trying to expand f(z) about z=z0, i.e.,
and we have z=z1 as the nearest point for which f(z) is not analytic. We
construct a circle C centered at z=z0 with radius
From the Cauchy integral formula,
0n
n
0n zzazf
010 zzzz
C 00C
zzzz
zdzf
i2
1
zz
zdzf
i2
1
zf
C 000 zzzz1zz
zdzf
i2
1
31. Here z is a point on C and z is any point interior to C. For |t| <1, we
note the identity
So we may write
which is our desired Taylor expansion, just as for real variable power
series, this expansion is unique for a given z0.
0
2
1
1
1
n
n
ttt
t
C n
n
n
zz
zdzfzz
i
zf
0
1
0
0
2
1
0
1
0
0
2
1
n C
n
n
zz
zdzf
zz
i
0
0
0
!n
n
n
n
zf
zz
32. Schwarz reflection principle
From the binomial expansion of for integer n (as an
assignment), it is easy to see, for real x0
Schwarz reflection principle:
If a function f(z) is (1) analytic over some region including the real axis
and (2) real when z is real, then
We expand f(z) about some point (nonsingular) point x0 on the real axis
because f(z) is analytic at z=x0.
Since f(z) is real when z is real, the n-th derivate must be real.
n
0xzzg
*n
0
**n
0
*
zgxzxzzg
**
zfzf
0
0
0
!n
n
n
n
xf
xzzf
*
0
0
0
**
!
zf
n
xf
xzzf
n
n
n
34. Drawing an imaginary contour line to convert our region into a simply
connected region, we apply Cauchy’s integral formula for C2 and C1,
with radii r2 and r1, and obtain
We let r2 r and r1 R, so for C1, while for C2, .
We expand two denominators as we did before
(Laurent Series)
zz
zdzf
i
zf
CC
21
2
1
00 zzzz 00 zzzz
21
000000 112
1
CC
zzzzzz
zdzf
zzzzzz
zdzf
i
zf
zdzfzz
zzizz
zdzf
zz
i
n
n C
n
n C
n
n
0
01
00
1
0
0
21
1
2
1
2
1
n
n
n zzazf 0
35. where
Here C may be any contour with the annular region
r < |z-z0| < R encircling z0 once in a counterclockwise sense.
Laurent Series need not to come from evaluation of
contour integrals. Other techniques such as ordinary series
expansion may provide the coefficients.
Numerous examples of Laurent series appear in the next chapter.
C
nn
zz
zdzf
i
a 1
0
2
1
36.
0
222
1
m
mnimn
i
n
er
drie
i
a
0
21 2
1
1
1
2
1
m
n
m
nn
z
zd
z
izz
zd
zi
a
1
1zzzf
0
1,22
2
1
m
mni
i
1-nfor0
-1nfor1
an
1
32
1
1
1
1
n
n
zzzz
zzz
The Laurent expansion becomes
Example:
(1) Find Taylor expansion ln(1+z) at point z
(2) find Laurent series of the function
If we employ the polar form
1
1
)1()1ln(
n
n
n
n
z
z
37. • Theorem
Suppose that a function f is analytic throughout an annular
domain R1< |z − z0| < R2, centered at z0 , and let C denote any
positively oriented simple closed contour around z0 and lying in
that domain. Then, at each point in the domain, f (z) has the
series representation
Laurent Series
0 1 0 2
0 1 0
( ) ( ) ,( | | )
( )
n n
n n
n n
b
f z a z z R z z R
z z
1
0
1 ( )
,( 0,1,2,...)
2 ( )
n n
C
f z dz
a n
i z z
1
0
1 ( )
,( 1,2,...)
2 ( )
n n
C
f z dz
b n
i z z
38. • Theorem (Cont’)
Laurent Series
0 1 0 2( ) ( ) ,( | | )n
n
n
f z c z z R z z R
0 1 0 2
0 1 0
( ) ( ) ,( | | )
( )
n n
n n
n n
b
f z a z z R z z R
z z
1
0
1 ( )
,( 0,1,2,...)
2 ( )
n n
C
f z dz
a n
i z z
1
0
1 ( )
,( 1,2,...)
2 ( )
n n
C
f z dz
b n
i z z
1
0
1 ( )
,( 0, 1, 2,...)
2 ( )
n n
C
f z dz
c n
i z z
1 1
0
0
( )
( )
nn
nn
n n
b
b z z
z z
, 1
, 0
n
n
n
b n
c
a n
39. • Laurent’s Theorem
If f is analytic throughout the disk |z-z0|<R2,
Laurent Series
0
0
( ) ( )n
n
n
f z a z z
1
01
0
1 ( ) 1
( ) ( ) ,( 1,2,...)
2 ( ) 2
n
n n
C C
f z dz
b z z f z dz n
i z z i
Analytic in the region |z-z0|<R2
0,( 1,2,...)nb n
( )
0
1
0
( )1 ( )
,( 0,1,2,...)
2 ( ) !
n
n n
C
f zf z dz
a n
i z z n
reduces to Taylor
Series about z0
0 1 0 2
0 1 0
( ) ( ) ,( | | )
( )
n n
n n
n n
b
f z a z z R z z R
z z
40. • Example 1
Replacing z by 1/z in the Maclaurin series expansion
We have the Laurent series representation
Examples
2 3
0
1 ...(| | )
! 1! 2! 3!
n
z
n
z z z z
e z
n
1/
2 3
0
1 1 1 1
1 ...(0 | | )
! 1! 2! 3!
z
n
n
e z
n z z z z
There is no positive powers of z, and all coefficients of the positive powers are zeros.
1
1 ( )
,( 1,2,...)
2 ( 0)
n n
C
f z dz
b n
i z
1/
1/
1 1 1
1 1
1
2 ( 0) 2
z
z
C C
e dz
b e dz
i z i
1/
2z
C
e dz i
where c is any positively oriented simple closed
contours around the origin
41. • Example 2
The function f(z)=1/(z-i)2 is already in the form of a
Laurent series, where z0=i,. That is
where c-2=1 and all of the other coefficients are zero.
Examples
2
1
( ) ,(0 | | )
( )
n
n
n
c z i z i
z i
3
0
1
,( 0, 1, 2,...)
2 ( )
n n
C
dz
c n
i z z
3
0, 2
2 , 2( )n
C
ndz
i nz i
where c is any positively oriented simple contour
around the point z0=i
42. Examples
Consider the following function
1 1 1
( )
( 1)( 2) 1 2
f z
z z z z
which has the two singular points z=1 and z=2, is analytic in the domains
1 :| | 1D z
3 : 2 | |D z
2 :1 | | 2D z
43. • Example 3
The representation in D1 is Maclaurin series.
Examples
1 1 1 1 1
( )
1 2 1 2 1 ( / 2)
f z
z z z z
1
1
0 0 0
( ) (2 1) ,(| | 1)
2
n
n n n
n
n n n
z
f z z z z
where |z|<1 and |z/2|<1
44. • Example 4
Because 1<|z|<2 when z is a point in D2, we know
Examples
1 1 1 1 1 1
( )
1 2 1 (1/ ) 2 1 ( / 2)
f z
z z z z z
where |1/z|<1 and |z/2|<1
1 1 1
0 0 1 0
1 1
( ) ,(1 | | 2)
2 2
n n
n n n n
n n n n
z z
f z z
z z
45. • Theorem 1
If a power series
converges when z = z1 (z1 ≠ z0), then it is absolutely
convergent at each point z in the open disk |z − z0| < R1
where R1 = |z1 − z0|
Some Useful Theorems
0
0
( )n
n
n
a z z
46. • Theorem
Suppose that a function f is analytic throughout a disk
|z − z0| < R0, centered at z0 and with radius R0. Then f (z)
has the power series representation
Taylor Series
0 0 0
0
( ) ( ) ,(| | )n
n
n
f z a z z z z R
( )
0( )
,( 0,1,2,...)
!
n
n
f z
a n
n
That is, series converges to f (z) when z
lies in the stated open disk.
1
0
1 ( )
2 ( )
n n
C
f z dz
a
i z z
Refer to pp.167
47. Proof the Taylor’s Theorem
( )
0 0
0
(0)
( ) ,(| | )
!
n
n
n
f
f z z z z R
n
Proof:
Let C0 denote and positively oriented circle |z|=r0, where r<r0<R0
Since f is analytic inside and on the circle C0 and since the
point z is interior to C0, the Cauchy integral formula holds
0
0
1 ( )
( ) , ,| |
2 C
f s ds
f z z z R
i s z
1 1 1 1 1
, ( / ),| | 1
1 ( / ) 1
w z s w
s z s z s s w
48. Proof the Taylor’s Theorem
1
1
0
1 1 1
( )
N
n N
n N
n
z z
s z s s z s
0
1 ( )
( )
2 C
f s ds
f z
i s z
0 0
1
1
0
1 ( ) 1 ( )
( )
2 2 ( )
N
n N
n N
n C C
f s ds f s ds
f z z z
i s i s z s
( )
(0)
!
n
f
n
Refer to pp.167
0
( )1
0
(0) ( )
( )
! 2 ( )
n NN
n
N
n C
f z f s ds
f z z
n i s z s
ρN
49. Proof the Taylor’s Theorem
0
( )
lim lim 0
2 ( )
N
N NN N
C
z f s ds
i s z s
( ) ( ) ( )1
0 0 0
(0) (0) (0)
( ) lim( ) 0
! ! !
n n nN
n n n
N
N
n n n
f f f
f z z z z
n n n
When
0
0
0 0
( ) | |
| | | | 2
2 ( ) 2 ( )
N N
N N N
C
z f s ds r M
r
i s z s r r r
Where M denotes the maximum value of |f(s)| on C0
0
0 0
| | ( )N
N
Mr r
r r r
lim 0N
N
0
( ) 1
r
r
50. Example
expand f(z) into a series involving powers of z.
We can not find a Maclaurin series for f(z) since it is not analytic at
z=0. But we do know that expansion
Hence, when 0<|z|<1
Examples
2 2
3 5 3 2 3 2
1 2 1 2(1 ) 1 1 1
( ) (2 )
1 1
z z
f z
z z z z z z
2 4 6 8
2
1
1 ...(| | 1)
1
z z z z z
z
2 4 6 8 3 5
3 3
1 1 1
( ) (2 1 ...) ...f z z z z z z z z
z z z
Negative powers
51. Residue theorem
Calculus of residues
Functions of a Complex Variable
Suppose an analytic function f (z) has an isolated singularity at z0. Consider a contour
integral enclosing z0 .
z0
)(sRe22)(
1,2)ln(
1,0
1
)(
)(
)()()(
01
1
'
'01
'
'
1
0
0
00
zfiiadzzf
niazza
n
n
zz
a
dzzza
dzzzadzzzadzzf
C
z
z
z
z
n
n
C
n
n
n
C
n
n
C
n
n
n
C
The coefficient a-1=Res f (z0) in the Laurent expansion is called the residue of f (z) at z = z0.
If the contour encloses multiple isolated
singularities, we have the residue theorem:
n
n
C
zfidzzf )(sRe2)(
z0 z1
Contour integral =2i ×Sum of the residues
at the enclosed singular points
52. Residue formula:
To find a residue, we need to do the Laurent expansion and pick up the coefficient a-1.
However, in many cases we have a useful residue formula
)()(lim
)!1(
1
)(sRe
)!1())(2()1)((lim)(lim
)(lim
)!1(
1
)()(lim
)!1(
1
:Proof
.)()(lim)(sRe
,polesimpleaforly,Particular
)()(lim
)!1(
1
)(sRe
,orderofpoleaFor
01
1
01
1
1
1
001
1
01
1
01
1
00
01
1
0
0
00
00
0
0
zfzz
dz
d
m
zfa
mazznmnmnazza
dz
d
zza
dz
d
m
zfzz
dz
d
m
zfzzzf
zfzz
dz
d
m
zf
m
m
m
m
zz
n
n
n
zz
mn
mn
nm
m
zz
mn
mn
nm
m
zz
m
m
m
zz
zz
m
m
m
zz
53.
.0,)()(lim
!
1
:tscoefficientheallfindway toaisethat therprovedactuallyWe
.)()(lim
)!1(
1
usgives1upPick.Also
.)()(lim
!
1
,)()()(
expansionTaylorbyanalytic,is)()(Because
)()()(
)()(
:#2MethodProof
0
01
1
1
00
0
0
0
00
0
0
0
0
kzfzz
dz
d
k
a
a
zfzz
dz
d
m
amkab
zfzz
dz
d
k
bzzbzfzz
zfzz
zzazfzz
zzazf
m
k
k
zz
mk
m
m
m
zz
mkk
m
k
k
zz
k
k
k
k
m
m
mn
mn
n
m
n
mn
n
54. Cauchy’s integral theorem and Cauchy’s integral formula revisited:
(in the view of the residue theorem):
.
!
)()(
)!11(
1
lim
isatresidueitsformula,residuethetoAccording
1.orderofpoleaisIt.
)(')()(
)'3
!
)(
22
)(
)(
)(
)3
)(2
)(
)('
)()(
)2
0)(Res2)()1
))((')()()(:functionAnalytic
0
)(
1
0
1
01)1(
1)1(
0
0
0
1
0
0
1
0
0
)(
1
00
1
01
0
0
0
0
0
0
0
0
000
0
0
0 n
zf
zz
zf
zz
dz
d
n
zz
n
zz
zf
zz
zf
zz
zf
n
zf
iiadz
zz
zf
zza
zz
zf
zifdz
zz
zf
zf
zz
zf
zz
zf
zfidzzf
zzzfzfzzazf
n
n
n
n
n
zz
nnn
n
n
C n
m
nm
mn
C
C
m
m
m
Evaluation of definite integrals -1
Calculus of residues
55.
2222
2
22
2
22
2
22
2
0
22
0
2
2
2
2
2
0
111
2
111
2
1
,
111
1
2
11
)1(1
)(
1111
)(Res)0(Res
.
)(
1
))((
1
lim)(Res
.
1
))((
1
lim)0(Res
circle.theofoutiscircle,in theis||||,1||
.1101/2,0poles,simple3haveWe
)1/2(
11
2
111
2//11
2//1
1.||andrealis,
sin1
sin
Example
aa
a
aa
a
i
ia
I
aa
a
a
a
i
a
a
izzz
zz
zz
z
zz
zff
zzz
z
zzzzz
z
zzzf
zzzzzzz
z
zf
zzzzzz
a
a
i
zaizzz
dz
aizzz
z
ia
dz
aizazz
z
iiz
dz
izza
izz
I
aa
a
d
I
zz
z
CCC
C
r=1
z+
z-
z0
56. Evaluation of definite integrals -2
Calculus of residues
II. Integrals along the whole real axis:
dxxf )(
Assumption 1: f (z) is analytic in the upper or lower half of the complex plane, except
isolated finite number of poles.
∩
R
Condition for closure on a semicircular path:
dzzfdzzfdzzfdzzfdzzfdxxf
RCR
R
RR
)(lim)(lim)()()(lim)(
.0,
1
~)(lim0lim)(lim
)(lim)(lim)(lim
1max
0
00
z
zfRfRdeRf
deiReRfdeiReRfdzzf
RR
i
R
ii
R
ii
RR
Assumption 2: when |z| , | f (z)| goes to zero faster than 1/|z|.
Then, plane.halfupperon the)(ofesiduesR2)(lim)( zfidzzfdxxf
CR