M4.1 Complex Analysis-Introdution

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 \(z=x+iy\)

Euler's Formula

We know the following series 

• \(e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...\)
• \(\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-+...\)
• \(\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-+...\)

If we multiply \(i\) to the above equation we get

• \(i \sin(x)=ix-\frac{ix^3}{3!}+\frac{ix^5}{5!}-+...\) 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 
• \(e^{ix}=1+ix-\frac{x^2}{2!}-\frac{ix^3}{3!}+\frac{x^4}{4!}+\frac{ix^5}{5!}-+... \,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\)

If we compare (1) and (2) we can conclude that

\(cos(x) + i sin(x) = e^{ix}\)

• \(e^{-ix} = \cos(x)-i \sin(x)\) Hence,
• \(\cos(x)=\frac{e^{ix}+e^{-ix}}{2}\)  & \(\sin(x)=\frac{e^{ix}-e^{-ix}}{2}\)
• De-Moivre Theorem: \((\cos(\theta)+i \sin(\theta))^n=\cos(n\theta)+i \sin(n\theta)\)

Complex Functions

Complex functions are mapping from complex numbers to complex numbers. Ex: \(f(z)=\frac{1}{z}=\frac{1}{x+iy}=\frac{x}{x^2+y^2}-i\frac{y}{x^2+y^2}=u(x,y)+iv(x,y)\)

Limit of a Complex Function

Let \(f(z) \) be a complex valued function, limit \(l\) as \(z\) tends to some point \(z_0\), there exists \(\epsilon, \delta\), when \(|z\to z_0|<\epsilon, |f(z)−l|<\delta\). i.e., \(lim_{z\to z_0}f(z)=l\)

Continuity of a complex function

If \(lim_{z\to z_0} f(z)=f(z_0) \) then \(f(z)\) is continuous at \(z=z_0\)
Example

Differentiablility of a complex function

Let \(\delta z=z−z_0\), as \(z\to z_0\) or \(\delta z\to 0, lim_{z\to z_0}\frac{f(z+\delta z)−f(z)}{\delta z}\) is the derivative of \(f\) at \(z_0\)

Analytic Functions

A complex function \(w=f(z) \) is said to be analytic at some point \(z_0\) if \(\frac{dw}{dz}=f′(z)=lim_{\delta z \to z_0} \frac{f(z+\delta z)−f(z)}{\delta z}\) exists and unique.