241-213 Computer Engineering Mathematics

1. 241-213Computer Engineering Mathematics 21

2. Chapter 21Complex Functions

3. OUTLINE • Limits • Continuity • Derivatives

4. Complex Functions • A complex function is a function that is defined for complex numbers in some set S and takes on complex values. • If C denotes the set of complex number, and • fis such a function, • then we write • This means, f(z) is a complex number for each z in S. • The set S is called the domainof f.

5. Example • Let S consists of all z with |z| < 1 • and define f(z) = z2 in S • then and f is a complex function. • Often we define a function by some explicit expression in z, for example, • In the absence of specifying S, we agree to allow all z which the expression for f(z) is defined • This function defines for all complex z except 2i and -2i

6. Limits

7. Limits

8. Example 21.1 Let

9. Limit Theorems • Many limit theorems from real calculus hold for complex functions as well • Suppose

10. Limits of Real vs Complex functions • Limits of real functions approach the point from 2 sides • Limits of complex functions approach the point from any directions

11. Continuity • Any polynomial is continuous for all z • Any rational function (quotient of polynomians) is continuous whenever its denominator is nonzero. • If f is continuous at z0, so is |f|

12. Continuity • If {zn} is a sequence of complex numbers and each f(zn) is defined • then {f(zn)} is also a complex sequence • For example, if f(z)= 2z2 and zn = 1/n then f(zn) = 2/n2 • We claim that {f(zn)} converges if {zn} does, when f is continuous • “Continuity preserves convergence of sequences”

13. THEOREM 21.1 • Let be continuous, and let {zn} be a sequence of complex numbers in S. If {zn} converges to a number w in S, then {f(zn)} converges to f(w) • A converse of Theorem 21.1 If f(zn)->f(w)for every sequence {zn} of points ofSconverging tow, thenfis continuous atw.

14. Bounded Function • If f is bounded if there is a disk about the origin containing all the numbers f(z) for z in S • A continuous function need not be bounded (look at f(z) = 1/z for z != 0)

15. THEOREM 21.2 • Let . Suppose S is compact and f is continuous on S. Then f is bounded.

16. THEOREM 21.3 • Let be continuous and suppose S is compact. Then there are numbers ziand z2 in Ssuch that, for allzinS

17. The Derivative of a Complex Function • The reason for having S open in this definition is to be sure that there is some open disk aboutz0 throughout which f(z) is defined

18. Example 21.2

19. Derivative Formulas • If n is a positive integer • and f(z) = zn • then f’(z) = nzn-1 • if f(z) = sin(z) then f’(z)= cos(z)

20. Derivative Formulas

21. Chain Rule

22. Example

23. THEOREM 21.4

24. Cauchy-Riemann Equations • We will derive a set of partial differential equations that must be satisfied by the real and imaginary parts of a differentiable complex function.

25. Example 21.4

26. Example 21.5

27. THEOREM 21.5 Cauchy-Riemann Equations

28. Proof

29. Example

30. Example

31. Note

32. Example 21.8

33. Example 21.8 (cont.)

34. Example 21.8 (cont.)

35. Example 21.8 (cont.)

36. THEOREM 21.6

37. THEOREM 21.7

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