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1. 1. Notes are adapted from D. R. Wilton, Dept. of ECE ECE 6382 Introduction to Complex Variables David R. Jackson 1 Fall 2022 Notes 1
2. 2. Some Applications of Complex Variables 2  Phasor-domain analysis in physics and engineering  Laplace and Fourier transforms  Series expansions (Taylor, Laurent)  Evaluation of integrals  Asymptotics (method of steepest descent)  Conformal Mapping (solution of Laplace’s equation)  Radiation physics (branch cuts, poles)
3. 3. Complex Arithmetic and Algebra A complex number z may be thought of simply as an ordered pair of real numbers (x, y) with rules for addition, multiplication, etc.       R , ( ) 1, e , Im cos sin i z x iy i j y z x x z z r i y re               (from figure) (Euler formula (not yet proven!)) 3 Note: In Euler's formula, the angle  must be in radians. Note: Usually we will use i to denote the square root of -1. However, we will often switch to using j when we are doing an engineering example. Argand diagram (polar form) Note: We can say that 1 i   But we need to be careful to properly interpret the square root (using the principal branch). This is what the radical sign usually denotes.       x y r  z z plane   arg z r z   
4. 4. Complex Arithmetic and Algebra 4 x y r  z z plane Note on phase angle (argument): The phase angle  is non-unique. We can add any multiple of 2 (360o) to it. This does not change x and y. Principal branch:       The most common choice for the “principal branch” is*: Note: Adding multiples of 2 to  will affect some functions, but not others. Examples:     f z z  noeffect     1/2 f z z  willeffect 2 p n      p       *e.g., the one that Matlab uses
5. 5. Complex Arithmetic and Algebra (cont.)         1 2 1 1 2 2 1 2 1 2 z z x iy x iy x x i y y          Addition / subtraction: 5 Geometrically, this works the same way and adding and subtracting two-dimensional vectors: “tip-to-tail rule” x y 1 z 2 z 1 2 z z  x y 1 z 2 z 1 2 z z  2 z 
6. 6. Complex Arithmetic and Algebra (cont.)                1 2 1 2 2 2 2 2 1 2 1 2 1 1 1 2 1 1 2 2 1 2 1 2 1 2 2 1 2 2 1 2 1 2 1 2 1 1 1 2 2 2 1 2 1 2 1 2 2 1 2 2 2 2 2 2 2 2 1 0 1 0 1 1 / / ( , ) i i i z z x iy x iy y x x y y i x y x y i i i z z re r e rr x iy x iy x x y x y e x iy z z x iy x x y y i y y x x y z z x y                                     Multiplication: Division:       1 2 1 2 2 2 1 2 2 1 1 2 1 2 2 2 2 / / i i i r z z re x r e e r y x x y                      6 Multiplication and division are easier in polar form! 1/ i i   Example :
7. 7. Complex Arithmetic and Algebra (cont.) 7  We can multiply and divide complex numbers. We cannot divide two-dimensional vectors. Important point: (We can, however, multiply two-dimensional vectors in two different ways, using the dot product and the cross product.)
8. 8. Complex Arithmetic and Algebra (cont.)         * * 2 2 * * i i z x iy z z z z r x y x iy x iy z z z r re re z z                 To see this : Conjugation: Magnitude : 8    y x r z r * z
9. 9. Euler’s Formula   2 3 0 2 3 0 2 4 3 5 0 1 2! 3! ! 1 2! 3! ! 1 ! 2! 4! 3! 5! cos sin n x n n z n n i n i x x x e x n x z x iy z z z e z n i e i n i e z                                                      Recall: Define extension to a complex variable ( ): (converges for all ) cos sin cos sin cos sin cos sin cos sin 2 2 i iz iz iz iz iz iz i e i e z i z e z i z e e e e z z i                       More generally, 9     cos cosh , sin sinh 2 2 2 z z z z z z e e e e e e iz z iz i i z i              Note: The variable  here is usually taken to be real, but it does not have to be. Leonhard Euler
10. 10. Application to Trigonometric Identities       2 2 2 2 2 2 cos2 sin 2 cos sin cos sin 2cos sin i i i e i e e i i                    Many trigonometric identities follow from a simple application of Euler's formula : On the other hand, Equatingreal andimaginary parts of t              1 2 1 2 1 2 2 2 1 2 1 2 1 1 2 2 1 2 1 2 1 2 1 cos2 cos sin sin 2 2cos sin cos sin cos sin cos sin cos cos sin sin sin cos cos i i i i e i e e e i i i                                             he two expressions yields identities: On the other hand, two       2 1 2 1 2 1 2 1 2 1 2 1 2 sin cos cos cos sin sin sin sin cos cos sin                   Equatingreal andimaginary parts yields: 10
11. 11. DeMoivre’s Theorem 11  2    2    x y z z             2 2 cos sin cos 2 sin 2 p p p p n n i n in n n i k i n kn n n z re r e r n i n n re r e r n kn i n kn k                                    (DeMoivre's Theorem) Note that for aninteger, the result is of how is measured ( anint ge e r) independent   cos sin p p n n r n i n z     Abraham de Moivre
12. 12. Roots of a Complex Number       2 2 1 1 1 1 2 2 cos sin , 0,1,2, 1 p p p k n n i i n k n n n n k k n n n n z re r e r i k n                                     roots 12 1 n w z        1 3 0: 8 2 cos sin 2 cos 30 sin 30 2 6 6 k i i i                                      3 2 1 2 i              1 3 1 3 3 , 2 2 1: 8 2 cos sin 2 cos 90 sin 90 2 , 6 3 6 3 4 4 2 : 8 2 cos sin 2 cos 210 sin 210 3 , 6 3 6 3 i k i i i i k i i i i                                                                                           2 2 6 3 2 1 3 1 3 2 2 8 8 2 2 cos sin , 0,1,2 6 3 6 3 k i i i i k k k i e e i k                                               E l xamp e:  In this case the results depend on how  is measured.
13. 13. Roots of a Complex Number (cont.)     1 3 2 2 2 1 1 1 1 3 , 8 2 , 3 p p p k n n k n n i i k i n i n n n n i i i i n n z re r e r e e z                                  "principal throot o branch" Note that the throot of can also be expressedin terms of the : th root of unity     1 1 2 2 2 2 1 cos sin , 0,1, , 1 k n n n i k n i k k e e i k n n n           f unity throot of unity where z x y 8i  u v w   1/3 1/3 8 w z i    Re Im 1 0   1 120   1 240   Cube root of unity (n = 3) 13 w u iv   Example (cont.)