ECE 6382 Integration in the Complex Plane David R. Jackson Notes are from D. R. Wilton, Dept. of ECE 1
Defining Line Integrals in the Complex Plane … … 2
Equivalence Between Complex and Real Line Integrals 3
Review of Line Integral Evaluation t t -a-a a t -a-a a … … … … 4
Review of Line Integral Evaluation (cont.) … … 5
Line Integral Example Although it is easier to use polar coordinates (see the next example), we use Cartesian coordinates to illustrate the previous Cartesian line integral form. Consider Hence 6
Line Integral Example (cont.) Consider 7
Line Integral Example (cont.) Consider 8
Line Integral Example (cont.) Consider Hence Note: By symmetry (compare z and –z ), we also have 9
Line Integral Example Consider Useful result and a special case of the “residue theorem” 10 Note: For n = -1, go back to integral.
Cauchy’s Theorem Consider 11 A simply-connected region means that there are no “holes” in the region. (Any closed path can be shrunk down to zero size.) Cauchy’s theorem:
Proof of Cauchy’s Theorem 12
Proof of Cauchy’s Theorem (cont.) 13 Some comments:
Cauchy’s Theorem (cont.) Consider 14
Extension of Cauchy’s Theorem to Multiply- Connected Regions 15
Cauchy’s Theorem, Revisited Consider Shrink the path down. 16
Fundamental Theorem of the Calculus of Complex Variables Calculus of Complex Variables Consider … … … 17 This is an extension of the same theorem in calculus (for real functions) to complex functions.
Fundamental Theorem of the Calculus of Complex Variables (cont.) Calculus of Complex Variables (cont.) Consider … … … 18 Starting assumptions:
Fundamental Theorem of the Calculus of Complex Variables (cont.) Calculus of Complex Variables (cont.) Consider … … … 19
Fundamental Theorem of the Calculus of Complex Variables (cont.) 20
Cauchy Integral Formula 21
Cauchy Integral Formula (cont.) 22
Cauchy Integral Formula (cont.) 23
Cauchy Integral Formula (cont.) Application: In graphical displays, one often wishes to determine if a point z 0 = (x,y) is hidden by a region (with boundary C ) in front of it, i.e. if in a 2-D projection, z 0 appears to fall inside or outside the region. (Just choose f(z) = 1.) 24 Summary
Derivative Formulas 25
Derivative Formulas (cont.) 26
Morera’s Theorem F will be analytic if we can prove its derivative exists! 27
Comparing Cauchy’s and Morera’s Theorems 28
Cauchy’s Inequality x y R Note: If a function is analytic within the circle, then it must have a convergent power (Taylor) series expansion within the circle (proven later). 29
Liouville’s Theorem Because it is analytic in the entire complex plane, f(z) will have a power (Taylor) series that converges everywhere. 30
The Fundamental Theorem of Algebra (due to Gauss*) Carl Friedrich Gauss As a corollary, an N th degree polynomial can be written in factored form as * The theorem was first proven Gauss’s doctoral dissertation in 1799 using an algebraic method. The present proof, based on Liouville’s theorem, was given by him later, in (his signature) 31
The Fundamental Theorem of Algebra (cont.) Note: Using the method of “polynomial division” we can construct the polynomial P N-1 (z) in terms of a n if we wish. An example is given on the next slide. 32
The Fundamental Theorem of Algebra (cont.) 33
Properties of Analytic Functions Analyticity Cauchy- Riemann conditions Path independence Morera’s theorem 34
Numerical Integration in the Complex Plane 35 Here we give some tips about numerically integrating in the complex plane. One way is to parameterize the integral: where so … …
Numerical Integration in the Complex Plane (cont.) 36 Each integral can be preformed in the usual way, using any convenient scheme for integrating functions of a real variable (Simpson’s rule, Gaussian Quadrature, Romberg method, etc.) If the function f is analytic, then the integral is path independent. We can choose a straight line path! Note: If the path is piecewise linear, we simply add up the results from each linear part of the path.
37 Note that if we sample uniformly in t, then we are really sampling uniformly along the line. We don’t have to sample uniformly, but we can if we wish. Numerical Integration in the Complex Plane (cont.)
38 Midpoint rule: Using Numerical Integration in the Complex Plane (cont.) N intervals t 0 1 tt t n mid tntn t1t1
39 “Complex Midpoint rule”: This is the same formula that we usually use for integrating a function along the real axis using the midpoint rule! Numerical Integration in the Complex Plane (cont.) N intervals where
40 “Complex Simpson’s rule”: N = number of segments = number of sample points = even number Numerical Integration in the Complex Plane (cont.) N intervals
41 “Complex Gaussian Quadrature”: Numerical Integration in the Complex Plane (cont.) x 1 = x 2 = x 3 = x 4 = x 5 = x 6 = w 1 = w 2 = w 3 = w 4 = w 5 = w 6 = (6-point Gaussian Quadrature) Sample points Weights Six sample points are used within each interval. N intervals Note: The Gaussian quadrature formulas are usually given for integrating a function over (-1,1). We translate these into integrating over interval n.
42 Numerical Integration in the Complex Plane (cont.) Here is an example of a piecewise linear path, used to calculate the electromagnetic field of a dipole source over the earth (Sommerfeld problem). Gaussian quadrature could be used on each of the linear segments of the path. Sommerfeld path