△x→0,△y=0 △u+iUau.O f(a)=lim △2-0△z△x-0 △x=0,4y→0 f(a)=lim Au t △u+i4u0u △z-0△z△y-0i△ ay 已a
Analytic Functions Elementary Functions Differentiability Analyticity ∆x → 0§∆y = 0 f 0 (z) = lim ∆z→0 ∆w ∆z = lim ∆x→0 ∆u + i∆v ∆x = ∂u ∂x + i∂v ∂x ∆x = 0§∆y → 0 f 0 (z) = lim ∆z→0 ∆w ∆z = lim ∆y→0 ∆u + i∆v i∆y = ∂v ∂y − i ∂u ∂y Cauchy-Riemann§ ∂u ∂x = ∂v ∂y ∂u ∂y = − ∂v ∂x C. S. Wu 1ù )Û¼ê
△x→0,△y=0 △u+iUau.O f(a)=lim △2-0△z△x-0 △x=0,4y→0 f(a)=lim Au t △u+i4u0u △z-0△z△y-0i△ ay auchy- Riemann方程 au a au
Analytic Functions Elementary Functions Differentiability Analyticity ∆x → 0§∆y = 0 f 0 (z) = lim ∆z→0 ∆w ∆z = lim ∆x→0 ∆u + i∆v ∆x = ∂u ∂x + i∂v ∂x ∆x = 0§∆y → 0 f 0 (z) = lim ∆z→0 ∆w ∆z = lim ∆y→0 ∆u + i∆v i∆y = ∂v ∂y − i ∂u ∂y Cauchy-Riemann§ ∂u ∂x = ∂v ∂y ∂u ∂y = − ∂v ∂x C. S. Wu 1ù )Û¼ê
Cauchy- demann方程 0_0 au ay a
Analytic Functions Elementary Functions Differentiability Analyticity Cauchy-Riemann§ ∂u ∂x = ∂v ∂y ∂u ∂y = − ∂v ∂x Cauchy-Riemann§´¼ê7^ Ø´¿©^ ±y²§ef(z) = u(x, y) + iv(x, y) ¢Ü u(x, y)ÚJÜv(x, y)þ1 § ÷v Cauchy-Riemann§§K¼êf(z) 1=o ê ∂u/∂x, ∂u/∂y, ∂v/∂xÚ∂v/∂y3 ëY C. S. Wu 1ù )Û¼ê
Cauchy- demann方程 0_0 au ay a o Cauchy- Rieman方程是函数可导的必要条件 但不是充分条件
Analytic Functions Elementary Functions Differentiability Analyticity Cauchy-Riemann§ ∂u ∂x = ∂v ∂y ∂u ∂y = − ∂v ∂x Cauchy-Riemann§´¼ê7^ Ø´¿©^ ±y²§ef(z) = u(x, y) + iv(x, y) ¢Ü u(x, y)ÚJÜv(x, y)þ1 § ÷v Cauchy-Riemann§§K¼êf(z) 1=o ê ∂u/∂x, ∂u/∂y, ∂v/∂xÚ∂v/∂y3 ëY C. S. Wu 1ù )Û¼ê
Cauchy- demann方程 0_0 au ay a Cauchy- Riemann方程是函数可导的必要条件 但不是充分条件 可以证明,若f(2)=0(x,0)+1(x,0)的实部 (x,0)和虚部(x,0)均可微,且满足 Cauchy-riemann方程,则函数()可导(朵
Analytic Functions Elementary Functions Differentiability Analyticity Cauchy-Riemann§ ∂u ∂x = ∂v ∂y ∂u ∂y = − ∂v ∂x Cauchy-Riemann§´¼ê7^ Ø´¿©^ ±y²§ef(z) = u(x, y) + iv(x, y) ¢Ü u(x, y)ÚJÜv(x, y)þ1 § ÷v Cauchy-Riemann§§K¼êf(z) 1=o ê ∂u/∂x, ∂u/∂y, ∂v/∂xÚ∂v/∂y3 ëY C. S. Wu 1ù )Û¼ê