Review of basic definitions & Key points
- The numbers are of the form \(x+iy\) are called as complex numbers, where \(x\) & \(y\) are Real numbers and \(i=\sqrt{-1}\).
- \(i=\sqrt{-1}\) is obtained from the equation \(x^2=-1\).
- \(1,i,i^2,i^3\) are the fourth roots of the unity and \(i^2=-1, i^3=-i, i^4=1\).
- Geometrically a complex number \(x+iy\) is a point in Cartesian plane as shown in the following figure
- From the figure we can say that \(r=\sqrt{x^2+y^2}\) and \(\theta=tan^{-1}{y/x}\) are called as modulus and amplitude of the complex number \(z\) denoted by \(|z|\) and \(amp(z)\) or \(arg(z)\) respectively
- \(z\bar=x-iy\) is the complex conjugate of the complex number \(z=x+iy\)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1jeTZ01S-Aw4mN91wehk1o0Kr6OD-3kBbN122qoYeShnxnM45Q-2v0hzvqAobfRZKEICqi3e7nJTdlUnRNOWlmsv5aXaQb5_mENtfhF8HSyJu241NJIWcHpV5eZZrjRZ2tojlnBwqf-w/s200/Complexnumber+geo.png)
Euler's Formula
We know the following series
- \(e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...\)
- \(\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-+...\)
- \(\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-+...\)
- \(i \sin(x)=ix-\frac{ix^3}{3!}+\frac{ix^5}{5!}-+...\) Now
- \(\cos(x)+i \sin(x)=1+ix-\frac{x^2}{2!}-\frac{ix^3}{3!}+\frac{x^4}{4!}+\frac{ix^5}{5!}-+... \,\,\,\,\,\,\,\,\,\, (1)\)
- Now
- \(e^{ix}=1+ix-\frac{x^2}{2!}-\frac{ix^3}{3!}+\frac{x^4}{4!}+\frac{ix^5}{5!}-+... \,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\)
\(cos(x) + i sin(x) = e^{ix}\)
- \(e^{-ix} = \cos(x)-i \sin(x)\) Hence,
- \(\cos(x)=\frac{e^{ix}+e^{-ix}}{2}\) & \(\sin(x)=\frac{e^{ix}-e^{-ix}}{2}\)
- De-Moivre Theorem: \((\cos(\theta)+i \sin(\theta))^n=\cos(n\theta)+i \sin(n\theta)\)
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