# how to find derivation of the following function

• November 1st 2012, 11:05 AM
flametag2
how to find derivation of the following function

$f(z) = e^{x^{2}+ y^{2}}[cos(2xy) + i sin(2xy)]$
• November 1st 2012, 02:46 PM
HallsofIvy
Re: how to find derivation of the following function
Technically the word is "differentiation", not "derivation".

I presume you know that if f(z)= f(x+ iy)= u(x, y)+ iv(x,y) then
$\frac{df}{dz}= \frac{\partial u}{\partial x}+ i \frac{\partial v}{\partial y}= \frac{\partial v}{\partial y}- i\frac{\partial v}{\partial y}$
(The fact that those two expressions on the right are equal is the "Cauchy- Rieman" equations. You can use either one.)

Here, of course, you have
$u(x,y)= e^{x^2+ y^2}cos(2xy)$
$v(x,y)= e^{x^2+ y^2}sin(2xy)$