# Homework Help: Complex Derivatives from frist principles

1. Sep 2, 2009

### NJunJie

1. The problem statement, all variables and given/known data

How to prove theres a derivative from first principles?

1/ [ z*sin(z)*cos(z) ]

z = complex = x + jy

It gets very completed.

2. Relevant equations

3. The attempt at a solution

2. Sep 2, 2009

### Dick

Indeed, it gets very complicated. Sure you CAN prove it. Work out u(x,y) and v(x,y) and show they satisfy Cauchy-Riemann. That's equivalent to complex differentiability. But that's just silly. It's a colossal amount of work. z, cos(z) and sin(z) are differentiable. Show they are differentiable from first principles. Then just use the quotient rule and the product rule. Haven't we talked about this in another thread??

3. Sep 2, 2009

### NJunJie

hi ya, thanks.
I've been trying and it just gets too complicated... :P
I've already proofed Cauchy Riemann for the denominator successfully - 2 pages long though.

Now i take the term alone and apply first prinicpes - just too COMPLEX. (well, i'll just state theorem as you mentioned - 'give up' heex).

Anyway, you've been a great help. :)

one more:-
sin (2z) / (z^15)

what is the location and nature of singularities here?