Chapter 2. Analysis Functions §2.2 Necessary and sufficient conditions of analytic functions. §2.1 The concept of analytic function §2.3 Elementary functions

§2.1 The concept of analytic function 1. Derivative ＆ differential Def have derivative at, if exists, the limit is called as derivative of at, denoted by, or Otherwise no derivative at.

Also Note: The forms are similar, however the derivative of the complex function is more complicated, strict with the differentiability.

Ex 1. Ex 2.

① let then (1) ② let then (2)

For Note 1. differentiable at, but not in any nbd of. Note 2., i.e. u & v i.e. Ref & Imf have continuous partial derivative of any order. exists.

Note 3. is domain,, exists Pf., ① let, then is real. ② let, then is pure imaginary.

· Ex.3 not differentiable on C. · Ex.4 not exist, Note 4. In general, f involve, not exist. Properties. 1. differentiable at continuous at.

2.The operations are same as real functions.

differentiable at or linear main part of

Def. exists, is the differential of at, denoted as when Derivative Differentiable

2. The concept of an analytic function Def. ① analytic at, if, differentiable at any. Also, is holomorphic at is regular at ② analytic on D (domain) if analytic at any. : singular point of, if is not analytic at

Ex for for, not exists not differentiable ∴ analytic on. Operations hold on.( ＋ － ×÷)

Let analytic on orC-R equation Ex. 1 §2.2 Necessary and sufficient conditions of analytic functions.

Ex.2 C-R equation holds only at is singular point. Note: exists. ① All partial derivatives are continuous & C-R equation exist. ② diff. at & C-R equation exist.

Ex. then but not exist. However, C-R equation holds at nbd of existanalytic

Theorem differentiable at any, and i.e. C-R equation Corollary.

Ex.2.2.3

Ex.2.2.5

Ex:2.2.6

Homework: P32-33: A1,A2,A3,A5,A6 *A7

§2.3 Elementary functions 1.Exponential function Def.

Property 1. i.e. e z is periodic function with elemental period - it’s one of the differences with real exponential function Property 2. however Ex. - another differences with real exponential function Property 3.

If,then,and so for some integer n. But because both lie in where the difference between the imaginary parts of any points is less than, we have. is one -to-one. Let with, we claim the equation has a solution Property 4.

The equationis then equivalent to the two equations This y is merely arg w, see Fig.1

2. Logarithmic function w is logarithmic function of z if denoted by Let multi-value, is principal value. when

∵ not continuous at (0,0) & arg(z) not continuous at (0,0) &. analytic on. Note: Equation of set Principal value equation. ??

Ex:

3. The Power Function For in general multi-value function

Ex:

4. Trigonometric function & Hyperbolic function Properties (1) Periodic odd even (2) (3)

(4) Def.

Hyperbolic function: Properties ①. sh z, ch z analytic on C, ②. sh z, ch z periodic with period ch z is even, sh z is odd. ③. unbounded

Ex:

5. Inverse Trigonometric & Hyperbolic

Inverse Hyperbolic

Homework: P33-34:A9-A13