# Cauchy-Riemman equations

• Oct 20th 2009, 09:10 PM
devouredelysium
Cauchy-Riemman equations
Ok. I am used to see written everywhere the following Cauchy-Rieamman equations for complex functions:
and

Now, today I saw this other notation
http://upload.wikimedia.org/math/0/9...98334feee9.png and I'd like to understand how it maps to the first. It must be pretty simple, I guess?

Thanks
• Oct 20th 2009, 09:30 PM
mr fantastic
Quote:

Originally Posted by devouredelysium
Ok. I am used to see written everywhere the following Cauchy-Rieamman equations for complex functions:
and

Now, today I saw this other notation
http://upload.wikimedia.org/math/0/9...98334feee9.png and I'd like to understand how it maps to the first. It must be pretty simple, I guess?

Thanks

f = u + iv.
• Oct 20th 2009, 09:34 PM
tonio
Quote:

Originally Posted by devouredelysium
Ok. I am used to see written everywhere the following Cauchy-Rieamman equations for complex functions:
and

Now, today I saw this other notation
http://upload.wikimedia.org/math/0/9...98334feee9.png and I'd like to understand how it maps to the first. It must be pretty simple, I guess?

Thanks

Yes it is, but only after one understands what's going on here: you have a complex variable function $f(z)$, but we can put $z=x+iy=(x,y)$, where $i=\sqrt{-1}$ and $(x,y)$ is the representation of a complex number in the complex plane.
Thus we can write $f(z)=f(x,y)=u(x,y)+iv(x,y)$ , with the usual division of the function in its real and imaginary functions $u , v$

Now, finally, applying the rule for partial derivatives of multivariable functions we get:

$i\frac{\partial f}{\partial x}=i\left(\frac{\partial u}{\partial x}+i\,\frac{\partial v}{\partial x}\right)=$ $\left(\frac{\partial u}{\partial y}+i\,\frac{\partial v}{\partial y}\right)=\frac{\partial f}{\partial y}$

Now just compare real and imaginary parts in both sides.

Tonio
• Oct 20th 2009, 09:38 PM
devouredelysium
Hmm was almoooost there. Now I get it! Thanks!