1. 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, , 10:46:42
Slide 1 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

2. 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, , 10:46:42
Slide 2 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

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

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

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

7. 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, , 10:46:42
Slide 7 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

9. 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, , 10:46:42
Slide 9 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

10. 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, , 10:46:42
Slide 10 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

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

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

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

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

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

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

18. 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, , 10:46:42
Slide 18 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

20. 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, , 10:46:42
Slide 20 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

22. 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, , 10:46:42
Slide 22 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

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

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

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

27. 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, , 10:46:42
Slide 27 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

29. 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, , 10:46:42
Slide 29 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

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

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

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

34. 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, , 10:46:42
Slide 34 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

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

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

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

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

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

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

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

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

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

45. 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, , 10:46:42
Slide 45 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

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

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

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

50. 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, , 10:46:42
Slide 50 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

52. 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, , 10:46:42
Slide 52 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

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

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

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

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

58. 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, , 10:46:42
Slide 58 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

59. 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, , 10:46:42
Slide 59 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

61. 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, , 10:46:42
Slide 61 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

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

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

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

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

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

68. 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, , 10:46:42
Slide 68 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

70. 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, , 10:46:42
Slide 70 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

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

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

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

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

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

77. 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, , 10:46:42
Slide 77 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

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

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

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

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

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

84. 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, , 10:46:42
Slide 84 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

85. 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, , 10:46:42
Slide 85 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

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

88. 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, , 10:46:42
Slide 88 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

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

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

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

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

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

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

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

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

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

98. 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, , 10:46:42
Slide 98 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

99. 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, , 10:46:42
Slide 99 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

100. 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, , 10:46:42
Slide 100 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

101. 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, , 10:46:42
Slide 101 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

102. 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, , 10:46:42
Slide 102 (of 102) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008