Complex Analysis

Question : Prove that tan iz=-i tanh z ? Answer : A complex number z is written as \[z=x+iy\] where x and y are real numbers. By the formula \[tan…

Question : What are real and imaginary parts of cosh z ? Solution : To find real and imaginary parts of cosh z where z is a complex number. A…

Question : What are real and imaginary parts of Sinh z ? Solution : To find real and imaginary parts of Sinh (z) where z is a complex number. A…

Question : Find real and imaginary parts of cos(z) Solution : Let z=x+iy be any complex numbers where x and y are real numbers. Now cos(z) can be written as…

Question : Prove that Re(z) and Im(z) are continuous functions of z Solution: Continuous function of z : A complex function f : A -> B is said to be…

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 \[\int_{C}{f(z)}dz=\int_{t=a}^{b}f\left(z\left(t\right)\right)\frac{dz(t)}{dt}dt\]…