講者： 許永昌 老師 1

Contents Preface Guide line of Ch6 and Ch7 Addition and Multiplication Complex Conjugation Functions of a complex variable Example: Electric Circuit Cauchy-Riemann Conditions for differentiable function. Derivatives of Elementary Functions. Analytic functions. 2

Preface Functions of complex variables can be used: 1. If f(z)=u+iv, and f(z) is an analytic function, we can get  2 u=  2 v=0 &  u  v=0 1. Electrostatic potential vs. E. 2. Create a curved coordinate. 2. Real numbers  Complex numbers 1. Fundamental theorem of algebra: 1. Any polynomial of order n has n (in general) complex zeros. 2. Real functions, infinite real series, and integrals usually can be generalized naturally to complex numbers simply by replacing a real variable x. 3. Propagation versus Evanescence 1. Helmholtz equation: (  2 +k 2 )u= . 4. Integrals: 1. Evaluating definite integrals. 2. Inverting power series 3. Infinite product representations of analytic functions 4. Obtaining solutions of differential equations for large values of some variable. 5. Investigating the stability of potentially oscillatory systems 6. Inverting integral transforms 5. Many physical quantities that were originally real become complex as a simple physical theory is made more general. 1. Index of refraction. 2. Impedance. 3

Guideline of Ch6 & Ch7 Basic complex algebra: +,- ,  Polar form Functions of a complex variable The Cauchy-Riemann condition for analytic functions Cauchy integral & Cauchy integral formula. Morera’s Theorem & Liouville’s Theorem Taylor expansion and Laurent expansion Conformal mapping Poles, branch points and branch cut lines. Residue Theorem Product expansion of entire function Count the number of poles and zeros. The leading term of an asymptotic expansion 4 *Reference: J. Bak, D.J. Newman, Complex Analysis

Complex algebra ( 請預讀 P319~P321) 5

Complex algebra (continue) Therefore, where x and y are real numbers. 6 x Re{z} y Im{z}  r

Exercise Find the roots of y(x)=x 2 +x+1. The polar form of 2+2i. e i  =cos  + isin , why? 7

Complex Conjugation ( 請預讀 P321~P323) 8

Exercise Example The Polar form of. 9

Triangle inequality ( 請預讀 P324) Please prove it. 10

Functions of a Complex Variable ( 請預 讀 P325) 11 x y u v z-plane w-plane

Multivalued functions ( 請預讀 P326) nth root: z=r  e i   e i2  m  m  Z z 1/n = r 1/n  e i  n  e i2  m/n, m=0,1, …, n-1. Logarithm: z=r  e i   e i2  m  m  Z lnz=lnr + i  (  +2  m),  m  Z Q: How to solve this problem? A: Cut line. (Will be discussed in Ch6.6) 12 The source of multivalue. Cut line ei2mei2m

Example ( 請預讀 P327) Electric Circuit: Based on Kirchhoff’s loop rule The steady-state solution: I=Re{I 0 e i  t }, Q=Re{I 0 /(i  ) e i  t }, dI/dt=Re{i  I 0 e i  t }, V=Re{V 0 e i  t }, We get I 0 {R+i  L  i/(  C)}=ZI 0 =V 0,  I 0 =V 0 /Z. Impedance: 13 R L C V 0 cos  t I Q -Q

Homework

Cauchy-Riemann Conditions ( 請預 讀 P331~P334) 15 Q: Can we say that if a complex function obeys this conditions, it is differentiable at z 0 ?

Cauchy-Riemann Conditions (continue) 16

Cauchy-Riemann Conditions (continue) A counterexample of a function whose f x and f y are not continuous but obey the Cauchy-Riemann Condition: Reference: J. Bak & D. J. Newman, Complex Analysis, P33. Non-differentiable at z=0: f y (0)=f x (0)=0 which obeys the Cauchy-Riemann Conditions. 17

Cauchy-Riemann Conditions (continue) ( 此條待考證 ) Where Therefore, 這就是為何我們在看複變時，看到的 f 都是 z 的函數，幾乎沒 看過 z* 的。 18

Cauchy-Riemann Conditions (final) Cauchy- Riemann Conditions fx(z)=ify(z). Differentiable at z: f ’(z) does exist 19 f x and f y are continuous at z.

Derivatives of elementary functions ( 請預讀 P334) We define the elementary functions by their Taylor expansion with x  z. In the convergent region, 20

Analytic Functions ( 請預讀 P335~P336) 21

Homework

Nouns 23