91Ó°ÊÓ

Concept used:

Formula from Cauchy Riemann theorem:

w = u + iv And z = x + iy, w = f(z) and z (x,y) , u (x,y) and u (x,y) .

Function is given as, w = sin z .

Now the real and imaginary part (u,v) is given as follows:

(u,v)

Here from Cauchy Riemann theorem w = u + iv and z = x + iy putting in above equation as follows:

u + iv =sin (x+yi) ...... (1)

u + iv = sin x cos yi + cos x sin yi

In previous sections concluded the following identities:

cos ir = cosh r sin ir

cos ir = i sinh r

So, by substitution in (ii) yields as follows:

u + iv = sin x cos yi + cos x sin yi

Obtain the values as follows:

u = sin x cosh y And v = cos x sinh y

If u,v constants and plot x,y

Hence, u = sin x cosh y And v = cos x sinh y