Complex Analysis

Question : Show that the function f(z)=sin(z)/z^4 at z=0 has a pole of order 3 at z=0 Solution : To Classify the singularity of f(z)=sin(z)/z^4 at z = 0 we…

Question : What is a zero of order m ? Show that f(z) = sin(z) − z has a zero of order 3 at z=0. Answer : Zero : A…

Question : Find real and imaginary parts of iz . Solution : We know that any complex number is written as \[z=x+iy\] where x and y are any real numbers…

Question : Find upper bound of the absolute value of the line integral of z^2 along a straight line segment C from 0 to 1 + i. Solution : To…

Question : Evaluate the line integral of 1/z along unit circle in counter clockwise direction starting from z = 1 Solution: Since a complex line integral is evaluated as \[\int_{C}{f(z)}dz=\int_{t=a}^{b}f\left(z\left(t\right)\right)\frac{dz(t)}{dt}dt\]…

Question : how to find singularities of cot z. Solution : Singularities of cot z Since we can write \[cot\ z\ =\frac{\cos{z}}{\sin{z}}=\frac{p\left(z\right)}{q\left(z\right)}\] Singularities of cot z are the zeros of…

Question : Prove that each singular point of tan z is a simple pole. Solution : Singularities of tan z Since we can write \[tan\ z\ =\frac{\sin{z}}{\cos{z}}=\frac{p\left(z\right)}{q\left(z\right)}\] Singularities of tan…

Question : how to Find zeros of tan z and cot z Solution : (a)zeros of tan z We can write \[tan\ z\ =\frac{\sin{z}}{\cos{z}}\] Now zeros of tan z are…

Question : Show that sin(z)/z has removable singularity at 0 Solution : Given function \[f(z)\ =\frac{sin\left(z\right)}{z}\] To that sin(z)/z has removable singularity at 0 First we shall find it’s Laurent…

Question : What is an isolated singularity? Explain with examples Solution : Isolated Singularity : A singular point z0 is called an isolated singularity of a function f(z) if f(z)…