Complex Analysis

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)…

Question : Find Real and Imaginary parts of the following complex numbersa) \[z = (2 + 3i)(−3 + 2i)\] b) \[z =\frac{4 + 2i}{3 − 4i}\]c) \[z = \bar{1 +…

Question : Using L Hospital s rule calculate lim(z→0) log^2((1+z)/z^2 Solution :  

Question : Using L Hospital s rule calculate lim_(z→0)(1 - cos z)/sin^2(z) Solution:

Question : Calculate the derivative of e^(z^2) Solution: Given function h(z) = e^(z^2) Put w=f(z) =z^2 and g(w)=e^w then h(z)=g(f(z)) Here  h(z) = g(f(z)) is the composition of functions  f(z)…

Question : Find the zeros of cos z Solution : zeros of cos z are obtained by solving cos z = 0 Put cos z = (e^(iz) + e^(−iz))/2 e^(iz)…