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Chapter 3. Elementary Functions

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 + i 0 .

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Chapter 3. Elementary Functions

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  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. It can be shown that (Ex.15) this is the only function satisfying conditions • 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. can be used to verify the additive property

  4. Ex : There are values of z such that

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

  6. Ex:

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

  8. A zero of a given function f (z) is a number z0such that f (z0)=0 • Since • And there are no other zeros since from (15)

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

  10. Frequently used identities

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

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

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

  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,

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

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

  18. 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

  19. 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.

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

  21. Solving for taking log on both sides. • Inverse Trigonometric and Hyperbolic Functions • write

  22. similarly, Example: But since

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