1. 1 Chapter 3. Elementary Functions Consider elementary functions studied in calculus and define corresponding functions of a complex variable. To be specific, define analytic functions of a complex variable z that reduce to the elementary functions in calculus when z = x+i0. 23. Exponential Function If f (z), is to reduce to when z=x i.e. for all real x, (1) It is natural to impose the following conditions: f is entire and for all z. (2) As shown in Ex.1 of sec.18 is differentiable everywhere in the complex plane and.

2. 2 It can be shown that (Ex.15) this is the only function satisfying conditions (1)and (2). And we write (3) when Euler’s Formula (5) since is positive for all x and since is always positive, for any complex number z.

3. 3 can be used to verify the additive property

4. 4 Ex : There are values of z such that

5. 5 24. Trigonometric Functions By Euler’s formula It is natural to define These two functions are entire since are entire.

6. 6 Ex:

7. 7 when y is real. in Exercise 7. unbounded

8. 8 A zero of a given function f (z) is a number z 0 such that f (z 0 )=0 Since And there are no other zeros since from (15)

9. 9 25. Hyperbolic Functions (3) (4)

10. 10 Frequently used identities

11. 11 From (4), sinhz and coshz are periodic with period

12. 12 26.The Logarithmic Function and Its Branches To solve Thus if we write

13. 13 Now, If z is a non-zero complex number,, then is any of, when Note that it is not always true that since has many values for a given z or, From (5),

14. 14 The principal value of log z is obtained from (2) when n=0 and is denoted by

15. 15 If we let denote any real number and restrict the values of in expression (4) to the interval then with components is single-valued and continuous in the domain. is also analytic,

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18. 18 27. Some Identities Involving Logarithms non-zero. complex numbers (1) Pf:

19. 19 Example: (A) (B) also Then (1) is satisfied when is chosen. has n distinct values which are nth routs of z Pf: Let

20. 20 28. Complex Exponents when, c is any complex number, is defined by where log z donates the multiple-valued log function. ( is already known to be valid when c=n and c=1/n ) Example 1: Powers of z are in general multi-valued. since

21. 21 If and is any real number, the branch of the log function is single-valued and analytic in the indicated domain. when that branch is used, is singled-valued and analytic in the same domain.

22. 22 In (1) now define the exponential function with base C. when a value of logc is specified, is an entire function of z. Example 3. The principal value of It is analytic in the domain

23. 23 29.Inverse Trigonometric and Hyperbolic Functions write Solving for taking log on both sides.

24. 24 Example: But since similarly,

25. 25