Question

Test whether the following functions are analytic exp(x).(cos y + i sin y)

Answer 1

The given f(z) is analytic function

i) δU/δx = eˣ (cos y) = δV/δy

ii) δU/δy = eˣ (-sin y) = - δV/δx

Hence the CR - equations are satisfied

Explanation:

Explanation:-

Analytic function:-

A point at which an analytic function ceases to posses a derivative is

called a singular point of the function.Thus the necessary and sufficient condition for a complex function f(Z) = U+i V is analytic in a region R are

δU/δx = δV/δy and δU/δy = -δV/δx ( C R equations)

Given f(Z) = eˣ(cos y+ i sin y)

Let U = eˣ(cos y) ....(i)

and

V = eˣ(sin y) ....(ii)

Differentiating equation(i) partially with respective to 'x'

δU/δx = eˣ (cos y)

Differentiating partially equation(i) with respective to 'y'

δU/δy = eˣ (-sin y)

Differentiating equation(ii) partially with respective to 'x'

δV/δx = eˣ (sin y)

Differentiating equation(ii) partially with respective to 'y'

δV/δy = eˣ (cos y)

Now

δU/δx = eˣ (cos y) = δV/δy

i) δU/δx = δV/δy

δU/δy = eˣ (-sin y) = - δV/δx

ii) δU/δy = - δV/δx

Hence the CR - equations are satisfied

There fore the given f(z) is analytic function