Q8P - Mathematical Methods in Physical Sciences

Question

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

w = cosh z

 

Step-by-Step Solution

Answer

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

Step 1: 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)   .

Step 2: 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

Step 3: 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.