Chapter 3 Integral of Complex Function §3.1 Definition and Properties §3.2 Cauchy Integral Theorem §3.3 Cauchy’s Integral Formula §3.4 Analytic and Harmonic.

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Presentation transcript:

Chapter 3 Integral of Complex Function §3.1 Definition and Properties §3.2 Cauchy Integral Theorem §3.3 Cauchy’s Integral Formula §3.4 Analytic and Harmonic Functions

Review

§3.1 Definition and Properties 1. Def. C smooth (or piecewise smooth) f : C→ C.

C —— path of integral f —— integrand Z —— integration variable the limit is called the integral of f along C, denoted by. If C is closed, we can write.

2. Evaluation

TH (3.1.2) Corollary: (3.1.4)

Ex.1 center of a circle radius

3. Properties ①②③ ④ ⑤ the length of C(Pf. P38) ⑥

① ② Ex.

③ Note: I independent of integration path.

Ex.3.1.2

Ex.3.1.3

Note: integration of f(z) dependent on integration path.

§3.2 Cauchy Integral Theorem —— continuous C-R equation TH.3.2.1(Cauchy TH)

TH P42 upper limit lower limit

TH P45

Def Properties ① G anti derivative of f on D G analytic on D ② G 1 and G 2 anti derivative of f on D G 1 =G 2 +constant on D. TH (Fundamental Theorem of Contour Integral)

Ex.3.2.1

Ex.3.2.2

Generalized Cauchy theorem in multi-connected domains TH D multi connected with multi closed contours Γ, f(z) analytic in D and on Γ.

- Deformation Theorem

Closed Deformation Theorem Ex Solutions:

Homework: P59-60: A1-A7

§3.3 Cauchy’s Integral Formula ＆ High Order Derivative Analysis:

1. Cauchy’s Integral Formula: (TH 3.3.1) Pf. ∵ f(z) continuous at z 0,

Note 1. f(z) on D depend on f(z) on C D: domain 2. f =g analytic on C f =g on D 3. f: → C analytic.

Ex:

Solution:

2. Existence of higher derivative TH f analytic on C & on D, Pf. n=1

Note. f(z) analytic on D f (n) (z) exist on D & analytic on D. n=1,2,… -the difference with real function

Ex:

§3.4 Analytic and Harmonic Function Def. real harmonic on D, if is called harmonic function on D. Def.u, v harmonic on D, v is harmonic conjugate of u if

Note. v harmonic conjugate of u u harmonic conjugate of v i.e. u+iv analytic on Dv+iu analytic on D Properties: TH.3.4.2

(3). v harmonic conjugate of u -u harmonic conjugate of v i.e. u+iv analytic on D v-iu analytic on D (4). v harmonic conjugate of u on D u harmonic conjugate of v on D u, v constants on D. (5). v 1,v 2 harmonic conjugate of u on D v 1 =v 2 + constant on D. Pf. u+iv 1 analytic on D, u+iv 2 analytic on D i(v 1 -v 2 ) analytic on Dv 1 -v 2 analytic on D (real) v 1 -v 2 =constant.

Question: Does u have a harmonic conjugate (u x =v y, u y =-v x ) on D? Does there exist an analytic f :D →C, u=Re f ? (v=Im f ) Ans. No in general. yes if D is simply connected.

D simply connected domain, u harmonic on D, Similarly,

Ex Solution:

Ex method2 method1

Homework: P60-61: A8-A17