Let's do the first part. The C-R equations state that
Given that we can use the C-R equations to find that
For here it is fairly obvious what v is...
Let be analytic. In each of the following, find given .
(1)
(2) (Figured this one out)
(3)
(4)
(5)
If someone could show me step by step how to do one or two of these problems I would appreciate it. I'm not sure where I am going with these since I don't have the answers. Thanks!
These are Cauchy-Riemann equation problems.
The method is the same each time, so if you can do (2) you should have been able to do the others.
Use the C–R equations. Here's how to do (1). If then . Integrate with respect to y to see that for some function f(x). (This function f(x) takes the place of the constant of integration when integrating , because x counts as a constant in the partial derivative.)
Now differentiate v with respect to x, to get . But , which tells us that . So f(x) is constant.
Conclusion: const.