<>复变函数积分的定义

代数式:z=x+iyz=x+iyz=x+iy
三角式:z=r(cosφ+isinφ)z=r(cos\varphi+isin\varphi)z=r(cosφ+isinφ)
指数式:z=reiφz=r e^{i\varphi}z=reiφ

<>复函数的几何意义



<>复数的运算

若 z1=r1eiφ1z_1=r_1e^{i\varphi_1}z1​=r1​eiφ1​和z2=r2eiφ2z_2=r_2e^{i\varphi_2}z2​=
r2​eiφ2​,则
积:z=z1+z2=r1r2ei(φ1+φ2)z=z_1+z_2=r_1r_2e^{i(\varphi_1+\varphi_2)}z=z1​+z2​=r1​
r2​ei(φ1​+φ2​)
商:z=z1z2=r1r2ei(φ1−φ2)
z=\frac{z_1}{z_2}=\frac{r_1}{r_2}e^{i(\varphi_1-\varphi_2)}z=z2​z1​​=r2​r1​​ei(φ
1​−φ2​)
若 z=reiφz=re^{i\varphi}z=reiφ,则
乘方:zn=rneinφz^n=r^ne^{in\varphi}zn=rneinφ
方根:zn=rnei(φn+2kπn)
\sqrt[n]{z}=\sqrt[n]{r}e^{i(\frac{\varphi}{n}+\frac{2k\pi}{n})}nz​=nr​ei(nφ​+n2k
π​)
对数:lnz=ln(reiφ)=ln∣r∣+iφlnz=ln(re^{i\varphi})=ln|r|+i\varphilnz=ln(reiφ)=ln∣r∣
+iφ
幂函数:zn=(reiφ)n=rneinφ=rn(cosnφ+isinnφ)
z^n=(re^{i\varphi})^{n}=r^ne^{in\varphi}=r^n(cosn\varphi+isinn\varphi)zn=(reiφ)n
=rneinφ=rn(cosnφ+isinnφ)
zn=enLnz=en(ln∣z∣+iArgz),k=0,±1,±2...
z^n=e^{nLnz}=e^{n(ln|z|+iArgz)},k=0,\pm1,\pm2...zn=enLnz=en(ln∣z∣+iArgz),k=0,±1,
±2...

<>共轭复数

若z=x+iy=reiφz=x+iy=re^{i\varphi}z=x+iy=reiφ,则zzz的共轭复数定义 z∗=x−iy=re−iφ
z^*=x-iy=re^{-i\varphi}z∗=x−iy=re−iφ为复数zzz的共轭复数,∣z∣2=zz∗\lvert z\rvert^2=zz^*∣z∣
2=zz∗。

<>欧拉公式

eiφ=∑n=0∞1n!(iφ)n=∑k=0∞i2k2k!φ2k+∑k=0∞i2k+12k+1!φ2k+1
e^{i\varphi}=\sum^{\infty}_{n=0}{\frac{1}{n!}(i\varphi)^n}=\sum^{\infty}_{k=0}{\frac{i^{2k}}{2k!}\varphi^{2k}}+\sum^{\infty}_{k=0}{\frac{i^{2k+1}}{2k+1!}\varphi^{2k+1}}
eiφ=∑n=0∞​n!1​(iφ)n=∑k=0∞​2k!i2k​φ2k+∑k=0∞​2k+1!i2k+1​φ2k+1
=∑k=0∞(−1)k2k!φ2k+∑k=0∞(−1)k2k+1!φ2k+1
=\sum^{\infty}_{k=0}{\frac{(-1)^{k}}{2k!}\varphi^{2k}}+\sum^{\infty}_{k=0}{\frac{(-1)^{k}}{2k+1!}\varphi^{2k+1}}
=∑k=0∞​2k!(−1)k​φ2k+∑k=0∞​2k+1!(−1)k​φ2k+1
=cosφ+isinφ=cos\varphi+isin\varphi=cosφ+isinφ

<>三角函数

sinφ=12i(eiφ−e−iφ)sin\varphi=\frac{1}{2i}(e^{i\varphi}-e^{-i\varphi})sinφ=2i1​(
eiφ−e−iφ)
cosφ=12(eiφ+e−iφ)cos\varphi=\frac{1}{2}(e^{i\varphi}+e^{-i\varphi})cosφ=21​(eiφ
+e−iφ)

<>复变函数的定义

若在复数平面上存在点集EEE,对EEE的每个点z=x+iyz=x+iyz=x+iy都有复数w=u+ivw=u+ivw=u+iv与之对应,则称www为zzz
的函数,zzz为www的变量,定义域为 EEE,记为:
w=f(z)=u(x,y)+iv(x,y),z∈Ew=f(z)=u(x,y)+iv(x,y), z\in Ew=f(z)=u(x,y)+iv(x,y),z∈E
也即:f:z=x+iy⟶w=u+ivf: z=x+iy\longrightarrow w=u+ivf:z=x+iy⟶w=u+iv
定义了一个复变函数实际上定义了两个相关联的实二元函数,因此复函数将具有独特的性质。
例如:
w=f(z)=z2=(x+iy)2=x2−y2+2ixyw=f(z)=z^2=(x+iy)^2=x^2-y^2+2ixyw=f(z)=z2=(x+iy)2=x
2−y2+2ixy
这样{u(x,y)=x2−y2v(x,y)=2xy \begin{cases} u(x,y)&=x^2-y^2\\ v(x,y)&=2xy
\end{cases}{u(x,y)v(x,y)​=x2−y2=2xy​

<>导数的定义

设w=f(z)w=f(z)w=f(z)是在区域BBB的定义的单值函数。若在BBB内的某点ZZZ,极限:
lim⁡△z→0△w△z=lim⁡△z→0f(z+△z)−f(z)△z\lim \limits_{\triangle
z\rightarrow0}\frac{\triangle w}{\triangle z}=\lim \limits_{\triangle
z\rightarrow0}\frac{f(z+\triangle z)-f(z)}{\triangle z}△z→0lim​△z△w​=△z→0lim​△zf
(z+△z)−f(z)​
存在,且与△z→0\triangle z\rightarrow0△z→0的方向无关,则称函数w=f(z)w=f(z)w=f(z)在zzz
点可导,称该极限为函数f(z)f(z)f(z)在zzz点的导数,记为f′(z)f'(z)f′(z)或dfdz\frac{df}{dz}dzdf​。
1、当△z\triangle z△z沿实轴xxx趋于000,即△y=0,△z=△x→0\triangle y=0,\triangle
z=\triangle x\rightarrow0△y=0,△z=△x→0时,有
lim⁡△z=△x→0f(z0+△z)−f(z0)△z==lim⁡△x→0u(x0+△x,y0)+iv(x0+△x,y0)−u(x0,y0)−iv(x0,y0
)△x=∂u∂x+i∂v∂x \begin{array}{ll} \lim \limits_{\triangle z=\triangle
x\rightarrow0}\frac{f(z_{0}+\triangle z)-f(z_0)}{\triangle z}&=\\
&=\lim\limits_{\triangle x\rightarrow0}\frac{u(x_0+\triangle
x,y_0)+iv(x_0+\triangle x,y_0)-u(x_0,y_0)-iv(x_0,y_0)}{\triangle x}\\
&=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x} \end{array}△z
=△x→0lim​△zf(z0​+△z)−f(z0​)​​==△x→0lim​△xu(x0​+△x,y0​)+iv(x0​+△x,y0​)−u(x0​,y0​)
−iv(x0​,y0​)​=∂x∂u​+i∂x∂v​​

2、当△z\triangle z△z沿虚轴yyy趋于000,即△x=0,△z=△y→0\triangle x=0,\triangle z=\triangle
y\rightarrow0△x=0,△z=△y→0时,有
lim⁡△z=△y→0f(z0+△z)−f(z0)△z==lim⁡△y→0u(x0,y0+△y)+iv(x0,y0+△y)−u(x0,y0)−iv(x0,y0
)i△y=∂v∂y−i∂u∂y \begin{array}{ll} \lim \limits_{\triangle z=\triangle
y\rightarrow0}\frac{f(z_{0}+\triangle z)-f(z_0)}{\triangle z}&=\\
&=\lim\limits_{\triangle y\rightarrow0}\frac{u(x_0,y_0+\triangle
y)+iv(x_0,y_0+\triangle y)-u(x_0,y_0)-iv(x_0,y_0)}{i\triangle y}\\
&=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y} \end{array}△z
=△y→0lim​△zf(z0​+△z)−f(z0​)​​==△y→0lim​i△yu(x0​,y0​+△y)+iv(x0​,y0​+△y)−u(x0​,y0​
)−iv(x0​,y0​)​=∂y∂v​−i∂y∂u​​
柯西黎曼方程(Cauchy-Riemann,C_R方程)是函数在一点可微的必要条件。

f′(z)=∂u∂x+i∂v∂x=∂v∂y−i∂u∂y=∂u∂x−i∂u∂y=∂v∂y+i∂v∂x \begin{array}{ll}
f'(z)&=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}\\
&=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y}\\
&=\frac{\partial u}{\partial x}-i\frac{\partial u}{\partial y}\\
&=\frac{\partial v}{\partial y}+i\frac{\partial v}{\partial x} \end{array}f′
(z)​=∂x∂u​+i∂x∂v​=∂y∂v​−i∂y∂u​=∂x∂u​−i∂y∂u​=∂y∂v​+i∂x∂v​​
也可写成:
{∂u∂x=∂v∂y∂v∂x=−∂u∂y \begin{cases} \frac{\partial u}{\partial
x}=\frac{\partial v}{\partial y}\\ \frac{\partial v}{\partial
x}=-\frac{\partial u}{\partial y}\\ \end{cases}{∂x∂u​=∂y∂v​∂x∂v​=−∂y∂u​​

{∂u∂r=1r∂v∂φ1r∂u∂φ=−∂v∂r \begin{cases} \frac{\partial u}{\partial
r}=\frac{1}{r}\frac{\partial v}{\partial \varphi}\\ \frac{1}{r}\frac{\partial
u}{\partial \varphi}=-\frac{\partial v}{\partial r} \end{cases}{∂r∂u​=r1​∂φ∂v​r1
​∂φ∂u​=−∂r∂v​​

<>解析函数的定义

若函数f(z)f(z)f(z)在z0z_0z0​点及其邻域上处处可导,则称f(z)f(z)f(z)字z0z_0z0​解析,在区域E上每点都解析,则称f(z)
f(z)f(z)在区域上的解析函数。

<>解析函数的性质

解析函数的实部和虚部通过柯西黎曼(C-R)方程相互联系:知其中一个函数,可求另一个函数。