Chapter 5 Integral Transforms and Complex Variable Functions Solving Applied Mathematical Problems with MATLAB CRC/Taylor & Francis Press Chinese version by Tsinghua University Press PPT by Wenbin Dong and Jun Peng, Northeastern University, PRC Proofread by Dingyu Xue & YangQuan Chen 星期四, 2008-4- 24, 10:46:42 Slide 1 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Chapter 5 Integral Transforms and Complex Variable Functions Laplace Transforms and Their Inverses Fourier Transforms and Their Inverses Other Integral Transforms Z Transforms and Their Inverses Solving Complex Variable Function Problems Chapter summary 星期四, 2008-4- 24, 10:46:42 Slide 2 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.1 Laplace Transforms and Their Inverses Definitions and properties Computer solutions to Laplace transform problems 星期四, 2008-4- 24, 10:46:42 Slide 3 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.1.1 Definitions and properties Mathematical definition of the one-sided Laplace transform Properties of Laplace transform: Linear property where and are scalars. 星期四, 2008-4- 24, 10:46:42 Slide 4 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Differentiation property Time-domain shift -domain property Differentiation property The nth order derivative 星期四, 2008-4- 24, 10:46:42 Slide 5 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Initial value property when initial values are 0, then, Integration property Zero initial conditions: the multiple integral: Initial value property 星期四, 2008-4- 24, 10:46:42 Slide 6 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Final value property Convolution property If has no pole with non-negative real part Convolution property where the convolution operator is defined as 星期四, 2008-4- 24, 10:46:42 Slide 7 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Inverse Laplace transform: Other properties Inverse Laplace transform: where is greater than the real part of the poles of function 星期四, 2008-4- 24, 10:46:42 Slide 8 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.1.2 Computer solutions to Laplace transform problems Problem solution procedures for Laplace transform: Define a symbolic variable such as t , and define time-domain function Directly call Laplace() function or 星期四, 2008-4- 24, 10:46:42 Slide 9 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Inverse Laplace transform: The syntax: Call pretty() or latex() function to further process the obtained symbolic results Inverse Laplace transform: The syntax: default variable is specify the domain variables and 星期四, 2008-4- 24, 10:46:42 Slide 10 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.1 Given perform its Laplace transform MATLAB solutions: Simplify the answer Result: 星期四, 2008-4- 24, 10:46:42 Slide 11 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.2 Given obtain its Laplace and inverse Laplace transforms 星期四, 2008-4- 24, 10:46:42 Slide 12 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.3 Solve the inverse Laplace transform: Direct solution: High precision numerical solution: 星期四, 2008-4- 24, 10:46:42 Slide 13 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.4 Given , derive the relationship between and . and comparison of the difference 星期四, 2008-4- 24, 10:46:42 Slide 14 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Consider the initial conditions: 星期四, 2008-4- 24, 10:46:42 Slide 15 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.5 Display the differentiation property of Laplace transform MATLAB solutions: Laplace transform of the eighth-order derivative: 星期四, 2008-4- 24, 10:46:42 Slide 16 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.6 For solve MATLAB solutions: Collect terms in the numerator polynomial: Results 星期四, 2008-4- 24, 10:46:42 Slide 17 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.2 Fourier Transforms and Their Inverses Definitions and properties Solving Fourier transform problems Fourier sine and cosine transforms Discrete Fourier sine, cosine transforms 星期四, 2008-4- 24, 10:46:42 Slide 18 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.2.1 Definitions and properties Definition of the Fourier transform: Definition of the inverse Fourier transform: 星期四, 2008-4- 24, 10:46:42 Slide 19 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Properties of the Fourier transform: Linear property: for scalars and Shift property Complex domain shift Differentiation property Fourier transform of the nth derivative 星期四, 2008-4- 24, 10:46:42 Slide 20 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Integration property Scaling property Convolution property the Fourier transform to the nth order integral Scaling property Convolution property 星期四, 2008-4- 24, 10:46:42 Slide 21 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.2.2 Solving Fourier transform problems The syntax of Fourier transform Fourier transform Transform the function of into a function of 星期四, 2008-4- 24, 10:46:42 Slide 22 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The syntax of Inverse Fourier transformation Transform the function of into a function of 星期四, 2008-4- 24, 10:46:42 Slide 23 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

In MATLAB the definition of Fourier transform In MATLAB the definition of inverse Fourier transform 星期四, 2008-4- 24, 10:46:42 Slide 24 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.7 Given , where compute the Fourier transform for Inverse Fourier transform: 星期四, 2008-4- 24, 10:46:42 Slide 25 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.8 Given with , compute the Fourier transform Simplified and reduced result: 星期四, 2008-4- 24, 10:46:42 Slide 26 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.9 Given that , using fourier() command and the direct integration method respectively to compute the Fourier transform. Using fourier() command: 星期四, 2008-4- 24, 10:46:42 Slide 27 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Using direct integration method: NOTE: Not all functions have their corresponding Fourier transforms 星期四, 2008-4- 24, 10:46:42 Slide 28 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.2.3 Fourier sinusoidal and cosine transforms The Fourier sinusoidal transform is defined as The Fourier cosine transform is defined as 星期四, 2008-4- 24, 10:46:42 Slide 29 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The inverse Fourier sinusoidal transform is defined as The inverse Fourier cosine transform is defined as 星期四, 2008-4- 24, 10:46:42 Slide 30 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.10 Given compute the Fourier cosine transforms MATLAB commands: 星期四, 2008-4- 24, 10:46:42 Slide 31 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Calling Maple functions Fourier sine transform Fourier cosine transform 星期四, 2008-4- 24, 10:46:42 Slide 32 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

inverse Fourier sine transform inverse Fourier cosine transform 星期四, 2008-4- 24, 10:46:42 Slide 33 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.11 Given , solve its Fourier cosine transform and inverse Fourier cosine transform using Maple functions. Fourier cosine transform: Inverse transform: 星期四, 2008-4- 24, 10:46:42 Slide 34 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.12 Given solve its Fourier cosine transform MATLAB solutions: 星期四, 2008-4- 24, 10:46:42 Slide 35 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.2.4 Discrete Fourier sine, cosine transforms Discrete Fourier sinusoidal transform Discrete Fourier cosine transform 星期四, 2008-4- 24, 10:46:42 Slide 36 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The inverse discrete Fourier sinusoidal transform The inverse discrete Fourier cosine transform 星期四, 2008-4- 24, 10:46:42 Slide 37 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.13 Given where ,compute the discrete Fourier sinusoidal transform. MATLAB solutions: 星期四, 2008-4- 24, 10:46:42 Slide 38 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.3 Other Integral Transforms Mellin transform Hankel transform solutions 星期四, 2008-4- 24, 10:46:42 Slide 39 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.3.1 Mellin transform The Mellin transform is defined as The inverse Mellin transform is defined as 星期四, 2008-4- 24, 10:46:42 Slide 40 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.14 Given , where , solve its Mellin transform. MATLAB solutions: Result: 星期四, 2008-4- 24, 10:46:42 Slide 41 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.15 Given solve its Mellin transforms for several and try to summarize the possible general case. MATLAB solutions for : 星期四, 2008-4- 24, 10:46:42 Slide 42 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

For general case, the Mellin transform equation is: The syntax Mellin transform Inverse Mellin transforms 星期四, 2008-4- 24, 10:46:42 Slide 43 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.16 Given compute Mellin transform with Maple, then perform inverse Mellin transform MATLAB solutions: 星期四, 2008-4- 24, 10:46:42 Slide 44 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.3.2 Hankel transform solutions The th order Hankel transform is defined as: where is a Bessel function The syntax of evaluating the Hankel transform 星期四, 2008-4- 24, 10:46:42 Slide 45 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The th order inverse Hankel transform is defined as: The syntax of evaluating inverse Hankel transform 星期四, 2008-4- 24, 10:46:42 Slide 46 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.17 Given , compute the zeroth-order Hankel transform MATLAB solutions: Result: 星期四, 2008-4- 24, 10:46:42 Slide 47 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Given , compute the zeroth-order Hankel transform MATLAB solutions: 星期四, 2008-4- 24, 10:46:42 Slide 48 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.4 Z Transforms and Their Inverses Definitions and properties of Z transforms and inverses Computations of Z transform 星期四, 2008-4- 24, 10:46:42 Slide 49 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.4.1 Definitions and properties of Z transforms and inverses The one-sided Z transform of a discrete sequence is defined as: The properties of Z transforms: Linear property: for scalars and 星期四, 2008-4- 24, 10:46:42 Slide 50 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Time domain translation property: Z-domain proportional property: Frequency domain differentiation property: Frequency domain integration property 星期四, 2008-4- 24, 10:46:42 Slide 51 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Initial value property Final value property: Where has no poles outside of the unit circle Convolution property: where the operator for discrete signals is defined as 星期四, 2008-4- 24, 10:46:42 Slide 52 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The inverse Z transform of is defined as 星期四, 2008-4- 24, 10:46:42 Slide 53 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.4.2 Computations of Z transform The syntax of Z transform The syntax of inverse Z transform 星期四, 2008-4- 24, 10:46:42 Slide 54 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.18 Given compute the Z transform MATLAB solutions: 星期四, 2008-4- 24, 10:46:42 Slide 55 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.19 Given determine the inverse Z transforms for variable , and summarize the general formula. Solve the problem for 星期四, 2008-4- 24, 10:46:42 Slide 56 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The general form for Z transform 星期四, 2008-4- 24, 10:46:42 Slide 57 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.5 Solving Complex Variable Function Problems Complex Variable Functions and Mapping Visualization Concept and computation of residues Partial fraction expansion for rational functions Inverse Laplace transform using partial fraction expansions Computing closed-path integrals 星期四, 2008-4- 24, 10:46:42 Slide 58 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.5.1 Complex Variable Functions and Mapping Visualization The syntax of generating polar grids The syntax of drawing the 3D complex mapping surface 星期四, 2008-4- 24, 10:46:42 Slide 59 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.20 Draw the 3D mapping surface of the complex variable function MATLAB solutions: 星期四, 2008-4- 24, 10:46:42 Slide 60 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.5.2 Concept and computation of residues If is a single pole, then the residue is defined as The syntax of single pole 星期四, 2008-4- 24, 10:46:42 Slide 61 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

If is a pole of multiplicity , the residue is defined as The syntax of -multiple 星期四, 2008-4- 24, 10:46:42 Slide 62 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.21 Compute the residues of the function MATLAB solutions: 星期四, 2008-4- 24, 10:46:42 Slide 63 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.22 Compute the residue of the function MATLAB solutions: Result: 星期四, 2008-4- 24, 10:46:42 Slide 64 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Change the value of 星期四, 2008-4- 24, 10:46:42 星期四, 2008-4- 24, 10:46:42 Slide 65 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.23 Compute the residues of the function The residue for : 星期四, 2008-4- 24, 10:46:42 Slide 66 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Try different values of : The general form of result: 星期四, 2008-4- 24, 10:46:42 Slide 67 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.5.3 Partial fraction expansion for rational functions The rational function where and are all constants. The syntax of getting the greatest common divisor (GCD) of two polynomials: 星期四, 2008-4- 24, 10:46:42 Slide 68 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.24 Given check whether the two polynomials are coprime MATLAB solutions: 星期四, 2008-4- 24, 10:46:42 Slide 69 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Reduce the two polynomials Suppose and are co-prime. Assume that the roots for , are not repeating. Then, Residues are computed as: 星期四, 2008-4- 24, 10:46:42 Slide 70 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Suppose is a k-multiple repeating root : Residues are obtained by: 星期四, 2008-4- 24, 10:46:42 Slide 71 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The syntax for the partial fraction expansion of a given rational function where and 星期四, 2008-4- 24, 10:46:42 Slide 72 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.25 Perform partial fractional expansion for MATLAB solutions: 星期四, 2008-4- 24, 10:46:42 Slide 73 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.26 Given where compute the partial fraction expansion MATLAB solutions: 星期四, 2008-4- 24, 10:46:42 Slide 74 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

MATLAB function for partial fraction expansion 星期四, 2008-4- 24, 10:46:42 Slide 75 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

continued from the previous slide The syntax of the function 星期四, 2008-4- 24, 10:46:42 Slide 76 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The mathematical expression: if the analytical solutions to the equation of denominator polynomial cannot be obtained, high-precision numerical solutions should be used instead. The mathematical expression: where is the numerical solution 星期四, 2008-4- 24, 10:46:42 Slide 77 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.27 Compute the partial fraction expansion to MATLAB command solutions: 星期四, 2008-4- 24, 10:46:42 Slide 78 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The solutions The residues, -17/8, 2, 1/8 星期四, 2008-4- 24, 10:46:42 星期四, 2008-4- 24, 10:46:42 Slide 79 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.28 Given where Using analytical method to perform partial fractional expansion 星期四, 2008-4- 24, 10:46:42 Slide 80 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

MATLAB command solutions: Results 星期四, 2008-4- 24, 10:46:42 Slide 81 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.29 Given MATLAB command solutions: Results 星期四, 2008-4- 24, 10:46:42 Slide 82 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.30 Given , where compute the partial fraction expansion Numerical method: 星期四, 2008-4- 24, 10:46:42 Slide 83 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Cannot be expressed easily, since analytical solutions does not exist Analytical method: Cannot be expressed easily, since analytical solutions does not exist 星期四, 2008-4- 24, 10:46:42 Slide 84 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.5.4 Inverse Laplace transform using partial fraction expansions Based on the equation: where and Construct MATLAB function to enhance the facilities provided in residue() 星期四, 2008-4- 24, 10:46:42 Slide 85 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Continued from the previous page The syntax of the function 星期四, 2008-4- 24, 10:46:42 Slide 86 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.31 Given , where compute the partial fraction expansion MATLAB command solutions: 星期四, 2008-4- 24, 10:46:42 Slide 87 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

5.5.5 Computing closed-path integrals Given the closed-path integral where is a closed-path in counterclockwise direction, then where the closed-path encircles poles, 星期四, 2008-4- 24, 10:46:42 Slide 88 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.32 Given compute the closed-path integral on , 星期四, 2008-4- 24, 10:46:42 Slide 89 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The partial fractional expansion is: As the poles and are single poles, and , are poles of multiplicity 3 the residues are computed as 7198933/144, -1349779/27, 11/9, 1/432 星期四, 2008-4- 24, 10:46:42 Slide 90 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The closed-path integral solution The path of the circle can be expressed as 星期四, 2008-4- 24, 10:46:42 Slide 91 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The MATLAB command solutions: 星期四, 2008-4- 24, 10:46:42 Slide 92 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Example 5.33 Compute where is (an anti-clockwise circle) MATLAB command solutions: 星期四, 2008-4- 24, 10:46:42 Slide 93 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Direct path integral method Compare with Direct path integral method 星期四, 2008-4- 24, 10:46:42 Slide 94 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

The path is changed to , the MATLAB command solutions: Direct path integral method 星期四, 2008-4- 24, 10:46:42 Slide 95 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Chapter Summary List of Relevant MATLAB Functions 星期四, 2008-4- 24, 10:46:42 Slide 96 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

星期四, 2008-4- 24, 10:46:42 Slide 97 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Laplace transform is a very important integral transform Laplace transform is a very important integral transform. In this chapter, we introduced the definition and properties of Laplace transform with an emphasis on how to use MATLAB to obtain the Laplace transform and inverse Laplace transform. 星期四, 2008-4- 24, 10:46:42 Slide 98 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

Fourier transform is another commonly used important integral transform. It can be used in frequency domain analysis of time domain signals. We introduced the definition and properties of Fourier transform first and then put emphasis on how to use MATLAB to obtain Fourier transform. We discussed several special types of Fourier transforms and their MATLAB solution methods such as sine, cosine Fourier transforms, sine, cosine Fourier transforms and their inverse transforms. We mainly focused on direct integration method and the MATLAB Maple calls. 星期四, 2008-4- 24, 10:46:42 Slide 99 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

This chapter also introduced two integral transforms not commonly seen in many books: Mellin transform and Hankel transform. These two transforms have not corresponding command in Symbolic Math Toolbox but they can be done using basic symbolic command. In addition, by MATLAB calling of Maple kernel, we can do Mellin transform and Hankel transform and their inverse transforms. 星期四, 2008-4- 24, 10:46:42 Slide 100 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

In discrete time dynamic system analysis, an important integral transform, Z transform, is introduced in this chapter. We gave the definition and properties of Z transform first. Then, we focused on how to use MATLAB to solve Z transform and inverse Z transform problems with numerous examples. 星期四, 2008-4- 24, 10:46:42 Slide 101 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

In complex variable functional theory, an important problem is how to obtain the singular point and its corresponding residue. This chapter introduced the concept and the solution methods in MATLAB based on partial fractional expansion approach. A new residue function is provided which is much more useful than the one provided in MATLAB. Additionally, we also discussed on the inverse Laplace transform using the partial fractional expansion method for rational functions. Finally, based on the residue theory, we provided a method on how to solve contour integral for complex variable functions. 星期四, 2008-4- 24, 10:46:42 Slide 102 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008