Merge branch 'main' of https://github.com/Trance-0/NoteNextra
This commit is contained in:
Zheyuan Wu
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So everything we have learned so far is to extend the real line to the complex plane.
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## Chapter 0 Calculus on Real values
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### Differentiation
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Let $f,g$ be function on real line and $c$ be a real number.
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$$
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\frac{d}{dx}(f+g)=f'+g'
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$$
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$$
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\frac{d}{dx}(cf)=cf'
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$$
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$$
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\frac{d}{dx}(fg)=f'g+fg'
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$$
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$$
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\frac{d}{dx}(f/g)=(f'g-fg')/g^2
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$$
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$$
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\frac{d}{dx}(f\circ g)=(f'\circ g)\frac{d}{dx}g
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$$
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$$
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\frac{d}{dx}x^n=nx^{n-1}
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$$
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$$
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\frac{d}{dx}e^x=e^x
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$$
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$$
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\frac{d}{dx}\ln x=\frac{1}{x}
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$$
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$$
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\frac{d}{dx}\sin x=\cos x
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$$
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$$
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\frac{d}{dx}\cos x=-\sin x
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$$
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$$
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\frac{d}{dx}\tan x=\sec^2 x
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$$
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$$
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\frac{d}{dx}\sec x=\sec x\tan x
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$$
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$$
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\frac{d}{dx}\csc x=-\csc x\cot x
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$$
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$$
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\frac{d}{dx}\sinh x=\cosh x
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$$
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$$
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\frac{d}{dx}\cosh x=\sinh x
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$$
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$$
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\frac{d}{dx}\tanh x=\operatorname{sech}^2 x
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$$
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$$
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\frac{d}{dx}\operatorname{sech} x=-\operatorname{sech}x\tanh x
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$$
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$$
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\frac{d}{dx}\operatorname{csch} x=-\operatorname{csch}x\coth x
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$$
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$$
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\frac{d}{dx}\coth x=-\operatorname{csch}^2 x
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$$
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$$
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\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}}
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$$
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$$
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\frac{d}{dx}\arccos x=-\frac{1}{\sqrt{1-x^2}}
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$$
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$$
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\frac{d}{dx}\arctan x=\frac{1}{1+x^2}
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$$
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$$
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\frac{d}{dx}\operatorname{arccot} x=-\frac{1}{1+x^2}
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$$
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$$
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\frac{d}{dx}\operatorname{arcsec} x=\frac{1}{x\sqrt{x^2-1}}
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$$
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$$
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\frac{d}{dx}\operatorname{arccsc} x=-\frac{1}{x\sqrt{x^2-1}}
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$$
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### Integration
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Let $f,g$ be function on real line and $c$ be a real number.
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$$
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\int (f+g)dx=\int fdx+\int gdx
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$$
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$$
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\int cfdx=c\int fdx
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$$
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$$
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\int e^x dx=e^x
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$$
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$$
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\int \ln x dx=x\ln x-x
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$$
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$$
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\int \frac{1}{x} dx=\ln|x|
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$$
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$$
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\int \sin x dx=-\cos x
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$$
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$$
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\int \cos x dx=\sin x
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$$
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$$
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\int \tan x dx=-\ln|\cos x|
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$$
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$$
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\int \cot x dx=\ln|\sin x|
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$$
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$$
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\int \sec x dx=\ln|\sec x+\tan x|
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$$
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$$
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\int \csc x dx=\ln|\csc x-\cot x|
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$$
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$$
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\int \sinh x dx=\cosh x
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$$
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$$
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\int \cosh x dx=\sinh x
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$$
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$$
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\int \tanh x dx=\ln|\cosh x|
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$$
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$$
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\int \coth x dx=\ln|\sinh x|
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$$
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$$
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\int \operatorname{sech} x dx=2\arctan(\tanh(x/2))
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$$
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$$
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\int \operatorname{csch} x dx=\ln|\coth x-\operatorname{csch} x|
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$$
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$$
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\int \operatorname{sech}^2 x dx=\tanh x
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$$
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$$
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\int \operatorname{csch}^2 x dx=-\coth x
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$$
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$$
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\int \frac{1}{1+x^2} dx=\arctan x
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$$
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$$
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\int \frac{1}{x^2+1} dx=\arctan x
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$$
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$$
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\int \frac{1}{x^2-1} dx=\frac{1}{2}\ln|\frac{x-1}{x+1}|
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$$
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$$
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\int \frac{1}{x^2-a^2} dx=\frac{1}{2a}\ln|\frac{x-a}{x+a}|
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$$
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$$
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\int \frac{1}{x^2+a^2} dx=\frac{1}{a}\arctan(\frac{x}{a})
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$$
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## Chapter 1 Complex Numbers
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### Definition of complex numbers
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@@ -18,6 +225,14 @@ $$
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(x_1 + y_1i) \cdot (x_2 + y_2i) = (x_1x_2 - y_1y_2) + (x_1y_2 + x_2y_1)i
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$$
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#### Modulus
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The modulus of a complex number $z = x + yi$ is defined as
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$$
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|z| = \sqrt{x^2 + y^2}=|z\overline{z}|
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$$
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### De Moivre's Formula
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Every complex number $z$ can be written as $z = r(\cos \theta + i \sin \theta)$, where $r$ is the magnitude of $z$ and $\theta$ is the argument of $z$.
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@@ -121,6 +336,12 @@ $$
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\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
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$$
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On the polar form, the Cauchy-Riemann equations are
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$$
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r\frac{\partial u}{\partial r} = \frac{\partial v}{\partial \theta}, \quad \frac{\partial u}{\partial \theta} = -r\frac{\partial v}{\partial r}
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$$
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### Holomorphic functions
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A function $f$ is said to be holomorphic on an open subset $G$ of $\mathbb{C}$ if $f$ is differentiable at every point of $G$.
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@@ -190,6 +411,21 @@ where $a,b,c,d$ are complex numbers and $ad-bc\neq 0$.
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### Properties of linear fractional transformations
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#### Matrix form
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A linear fractional transformation can be written as a matrix multiplication:
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$$
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\phi(z) = \begin{bmatrix}
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a & b\\
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c & d\\
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\end{bmatrix}
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\begin{bmatrix}
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z\\
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1\\
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\end{bmatrix}
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$$
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#### Conformality
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A linear fractional transformation is conformal.
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So if $z_1,z_2,z_3$, $w_1,w_2,w_3$ are distinct points in the complex plane, then there exists a unique linear fractional transformation $\phi$ such that $\phi(z_i)=w_i$ for $i=1,2,3$.
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#### Inversion
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#### Factorization
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Every linear fractional transformation can be written as a composition of homothetic mappings, translations, inversions, and multiplications.
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If $\phi(z)=\frac{az+b}{cz+d}$, then
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$$
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\phi(z) = \frac{b-ad/c}{cz+d}+\frac{a}{c}
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$$
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#### Clircle
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A linear-fractional transformation maps circles and lines to circles and lines.
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## Chapter 4 Elementary Functions
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### Exponential function
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The exponential function is defined as
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$$
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e^z = \sum_{n=0}^\infty \frac{z^n}{n!}
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$$
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Let $z=x+iy$, then
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$$
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\begin{aligned}
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e^z &= e^{x+iy}\\
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&= e^x e^{iy}\\
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&= e^x\sum_{n=0}^\infty \frac{(iy)^n}{n!}\\
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&= e^x\sum_{n=0}^\infty \frac{(-1)^n y^{2n}}{(2n)!} + i \sum_{n=0}^\infty \frac{(-1)^n y^{2n+1}}{(2n+1)!}\\
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&= e^x(\cos y + i\sin y)\\
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\end{aligned}
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$$
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So we can rewrite the polar form of a complex number as
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$$
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z = r(\cos \theta + i\sin \theta) = re^{i\theta}
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$$
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#### $e^x$ is holomorphic
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Let $f(z)=e^z$, then $u(x,y)=e^x\cos y$, $v(x,y)=e^x\sin y$.
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$$
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\frac{\partial u}{\partial x} = e^x\cos y = \frac{\partial v}{\partial y}\\
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\frac{\partial u}{\partial y} = -e^x\sin y = -\frac{\partial v}{\partial x}
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$$
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### Trigonometric functions
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$$
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\sin z = \frac{e^{iz}-e^{-iz}}{2i}, \quad \cos z = \frac{e^{iz}+e^{-iz}}{2}, \quad \tan z = \frac{\sin z}{\cos z}
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$$
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$$
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\sec z = \frac{1}{\cos z}, \quad \csc z = \frac{1}{\sin z}, \quad \cot z = \frac{1}{\tan z}
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$$
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#### Hyperbolic functions
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$$
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\sinh z = \frac{e^z-e^{-z}}{2}, \quad \cosh z = \frac{e^z+e^{-z}}{2}, \quad \tanh z = \frac{\sinh z}{\cosh z}
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$$
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$$
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\operatorname{sech} z = \frac{1}{\cosh z}, \quad \operatorname{csch} z = \frac{1}{\sinh z}, \quad \operatorname{coth} z = \frac{1}{\tanh z}
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$$
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### Logarithmic function
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The logarithmic function is defined as
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$$
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\ln z=\{w\in\mathbb{C}: e^w=z\}
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$$
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#### Properties of the logarithmic function
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Let $z=x+iy$, then
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$$
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|e^z|=\sqrt{e^x(\cos y)^2+(\sin y)^2}=e^x
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$$
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So we have
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$$
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\log z = \ln |z| + i\arg z
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$$
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### Power function
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### Inverse trigonometric functions
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@@ -238,8 +552,21 @@ So if $z_1,z_2,z_3$, $w_1,w_2,w_3$ are distinct points in the complex plane, the
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### Definition of power series
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A power series is a series of the form
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$$
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\sum_{n=0}^\infty a_n (z-z_0)^n
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$$
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### Properties of power series
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#### Geometric series
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$$
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\sum_{n=0}^\infty z^n = \frac{1}{1-z}, \quad |z|<1
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$$
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### Radius/Region of convergence
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### Cauchy-Hadamard Theorem
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