Q7P - 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   constant should be orthogonal to the curves v =   constant . w = sin z

Step-by-Step Solution

Answer

The answer is 

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

                                                                                                       

Step 1: Cauchy Riemann Theorem

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)  .

Step 2: Use Cauchy Riemann Theorem

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 

Step 3: Get the values of (u,v)

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