CHAPTER 8 FUNCTIONS OF A COMPLEX VARIABLE

1. CHAPTER 8 FUNCTIONS OF A COMPLEXVARIABLE ANALYTIC PROPERTIES CALCULUS OF RESIDUES CONFORMAL MAPPING 呂學士 and 楊燿州 The imaginary numbers are a wonderful flight of God's spirit, they are almost an amphibian between being and not being. ----Gottfried Wilhelm Von Leibnitz, 1702.

2. 8.0 Introduction • We turn now to a study of functions of a complex variable. To indicate, at least partly, why complex variables are important, we mention briefly two areas of application. • For many pairs of functions u and v, both u and v satisfy Laplace's equation

3. Hence either u or v may be used to described a two-dimensional electrostatic potential. The other function which gives a family of curves orthogonal to those of the first function, may then be used to describe the electric field E. A similar situation holds for the hydrodynamics of an ideal fluid in irrotational motion. The function u might describe the velocity potential, whereas the function v would then be the stream function. In many cases in which the functions u and v are unknown mapping or transforming in the complex plane permits us to create a coordinate system tailored to the particular problem.

5. Integrals in the complex plane have a wide variety of useful applications. • Evaluating definite integrals. • Obtaining solutions of differential equations for large values the variable (asymptotic solutions). • Investigating the stability of potentially oscillatory systems.

6. 8.1 Complex Algebra A complex number is nothing more than an ordered pair of two ordinary real numbers, (a, b) or a+ib, in which i is . Similarly, a complex variable is an ordered pair of two real variables, z = (x, y) = x + iy.

7. x is called the real part of z and y is called the imaginary part of z. In MAXIMA command, this can be done as follows:

8. The operations of complex numbers are straight forward. Suppose z1= a+ib, z2=c+id, then • Additon, Substraction: • 2. Multiplication: • 3. Division: • 4. Conjugate:

9. For examples:

10. The expand of d did not give us the normal expression for the result after division, so we use realpart and imagpart commands to get the final results. • However, there is a more convenient command to show the real part and imaginary part simultaneously. This command is "rectform" as we will discuss later.

11. Complex Plane: It is frequently convenient to employ a graphical representation of the complex variable. By plotting x, the real part of z, as the abscissa and y, the imaginary part of z, as the ordinate, we have the complex plane or Argand plane shown in Fig. 8.1. If we assign specific values to x and y, then z corresponds to a point (x, y) in the plane. Fig. 8.1 Complex plane---Argand diagram

12. Further, from Fig. 8.1, we may write where and so,

13. In the last representation, r is called the modulus or magnitude of z, and the angle is labeled as the argument or phase of z. In MAXIMA command, cabs(exp) returns the complex absolute value or modulus of exp, and carg(exp) returns the argument or complex phase angle of exp.

14. Therefore, complex numbers, variables, or functions have two representations: one is in rectangular form and the other is in polar form. In MAXIMA, this is done by rectform (exp) : returns an expression of the form a + bi, where a and b are purely real. polarform (exp) : returns where r and q are purely real.

15. For examples:

16. From our complex variablez, complex functions f(z) or w(z) may be constructed. All of the elementary functions of real variables may be extended into the complex plane  replacing the real variable x by complex variable z. These complex functions may then resolved into real and imaginary parts w(z) = u (x, y) + iv(x,y) in which the separate functions u(x,y) and v(x,y) are pure real.

17. For example, if f(z)=z2, we have This can be checked by MAXIMA as follows:

18. We know from De Moivre' s formula that Therefore

19. which means Moreover

21. 8.2 Cauchy-Riemann Conditions • Having established complex functions of a complex variable, we now proceed to differentiate them. The derivative of f(z), like that of a real function, is defined by • Examples of functions which are not analytical: • absolute value function, • complex conjugate function • piecewise-defined function

22. The existance of df/dz is guaranteed by Cauchy-Riemann conditions. If f(z) is differentiable, the function f(z) is called analytic. Suppose that f(z) = u(x,y) +iv(x,y), then the Cauchy-Riemann conditions are • It can also be proved that if f(z) is analytic, then Cauchy-Riemann conditions hold. Therefore, f(z) is analytic <=> Cauchy-Riemann conditions hold.

23. Obviously, f(z) is analytic and Cauchy-Riemann conditions hold.

24. It is well worth noting that the Cauchy-Riemann conditions guarantee that the curve u=c1 will be orthogonal to the curves v=c2. This is fundamental in application to potential problems in a variety of areas of physics. If u=c1is an equipotential line (surface), v=c2 is a line of electric force and vice versa. • Example: Find the analytic function • f(z)=u(x,y)+iv(x,y) if u(x,y)=x2-y2

25. Solution: From the above results, we conclude v(x,y)=2xy.

26. In fact, we find that the function f(z) is z2 as we have seen previously. The plot of v=c2 will be orthogonal to the plot of u=c1 as we just discussed and can be illustrated below:

27. Fig. 8.2 3D plots u(x,y)=x2-y2 and v(x,y)=2xy. Note that v is obtained from u by Cauchy-Riemann conditions and its contour is orthogonal to that of u.

29. 8.3 Cauchy Integral Formula Consider a function f(z) that is analytic on a closed contour C and within the interior region bounded by C. Cauchy proved that in which z0 is some point in the interior region bounded by C. From this, it can also be proved that

30. Example: Find Solution: Therefore, I=4pi

31. Example: Find Solution: Therefore the answer is

32. 8.4 Laurent Expansion A complex function f(z) can be expressed as follows: This called Laurent expansion and the series is called Laurent series.

33. Example: Find the Laurent series of at z=1/2 up to 2nd order of z.

34. 8.5 Calculus of Residues Residue theorem. If the Laurent expansion of a function is integrated term by term by using a closed contour that encircles one isolated singular point zo once in a counterclockwise sense, we obtain However, if n = -1,

35. Summarizing the above two equations, we have The constant a-1 , the coefficient of ( z-z0 )-1 in the Laurent expansion, is called the residue of f(z) at z=z0. If there are multiple residues other than one, then we have

36. Example: Find the residue of s/( s2+a2 ) at s = ai and sin(ax)/x4at x = 0. Solution:

37. Example: Find the integral value of Solution: Therefore the integral value is

38. The calculus of residue is useful in evaluating a wide variety of definite integrals in both physical and purely mathematical problems. However, most of the problems can now be done directly by MAXIMA without resorting to the help of calculus of residues. Hence we skip this kind of application.