Question

To find u and vas a function of xand y & plot the graph and show curve u = constant should be orthogonal to the curves v =constant.

w = cosh z

Short answer

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

Step-by-step solution

Cauchy Riemann Theorem

Concept used:

Formula from Cauchy Riemann theorem:

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

Use Cauchy Riemann theorem

Function is given as. w = cosh z ...... (1)

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

Here from Cauchy Riemann theorem w = u + iv and z =x + iy putting in equation.

u + iv = cosh (x+yi) ...... (2)

u + iv = cosh x cosh yi + sinh x sinh yi

In previous section concluded the following identities.

cosh ir = cos r

sinh ir = i sin r

Substitution of (ii) yields

Substitute yields as follows:

u + iv = cosh x cosh yi + sinh x sinh yi

u + iv = cosh x cos y + i sinh x sin y

So, get the value (u,v) which is given as follows:

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

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