presentation.pptx
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.)