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Friday, 20 February 2015

Complex Number

A complex number x+yi is defined to be the point in the Argand diagram with coordinates (x, y).

Properties of complex number



As can be seen in the figure, x=r\cos\theta,y=r\sin\theta.
Then, the polar form will be r(\cos\theta+i\sin\theta)=rcis\:\theta, where cis\:\theta=\cos\theta+i\sin\theta.
Modulus: |z|=r=+\sqrt{x^2+y^2}
\tan\theta=\frac{y}{x}
Argument: arg\:z=\theta

\begin{align}zw&=r(\cos\theta+i\sin\theta)r'(\cos\phi+i\sin\phi)\\ &=rr'[(\cos\theta\cos\phi-\sin\theta\sin\phi)+i(\cos\theta\sin\phi-\sin\theta\cos\phi)]\\ &=rr'[\cos(\theta+\phi)+i\sin(\theta+\phi)]\end{align}
|zw|=|z||w|
Arguments add upon multiplication.

z^n=r^n(\cos\theta+i\sin\theta)^n=r^n(\cos n\theta+i\sin n\theta)
This is due to De Moive's Theorem.

Root of unity (applied in number theory, the theory of group characters, the discrete Fourier transform, etc)
A complex number that gives 1 when raised to some integer power n.
z^n=1

\epsilon=\cos\frac{2\pi}{n}+i\sin\frac{2\pi}{n}\\ \epsilon^k=\cos\frac{2\pi k}{n}+i\sin\frac{2\pi k}{n}\\ \epsilon^{n-1}=\cos\frac{2\pi (n-1)}{n}+i\sin\frac{2\pi (n-1)}{n}\\ \epsilon^{n}=1\\ |\epsilon|=1 \Rightarrow \epsilon^k\:\:\text{lie on the unit circle.}



Complex Exponential Taylor Series
e^z=\sum_{n=0}^\infty \dfrac{z^n}{n!}\\ \cos z=\sum_{n=0}^\infty \dfrac{(-1)^nz^{2n}}{(2n)!}\\ \sin z=\sum_{n=0}^\infty \dfrac{(-1)^n z^{2n+1}}{(2n+1)!}

Direct multiplication of series shows \large e^{z+w}=e^z\cdot e^w, thus \large e^z=e^{x+yi}=e^x\cdot e^{iy}.
e^{iy}=\sum_{n=0}^\infty \dfrac{(iy)^n}{n!}=\sum_{n=0}^\infty \dfrac{(-1)^nz^{2n}}{(2n)!}+\sum_{n=0}^\infty \dfrac{(-1)^n z^{2n+1}}{(2n+1)!}
Sidenote: e^{2\pi i}=1,e^{\pi i}=-1,e^{\frac{\pi i}{2}}=i,e^{\frac{3\pi i}{2}}=-i,e^{\frac{2\pi i}{8}}=\sqrt{i}=\dfrac{\sqrt{2}}{2}+i\dfrac{\sqrt{2}}{2}

More to explore:
http://www.math.vt.edu/people/dlr/m2k_bas_cmpext.pdf

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