Ok. I am used to see written everywhere the following Cauchy-Rieamman equations for complex functions:
and
Now, today I saw this other notation
and I'd like to understand how it maps to the first. It must be pretty simple, I guess?
Thanks
Ok. I am used to see written everywhere the following Cauchy-Rieamman equations for complex functions:
and
Now, today I saw this other notation
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 $\displaystyle f(z)$, but we can put $\displaystyle z=x+iy=(x,y)$, where $\displaystyle i=\sqrt{-1}$ and $\displaystyle (x,y)$ is the representation of a complex number in the complex plane.
Thus we can write $\displaystyle f(z)=f(x,y)=u(x,y)+iv(x,y)$ , with the usual division of the function in its real and imaginary functions $\displaystyle u , v$
Now, finally, applying the rule for partial derivatives of multivariable functions we get:
$\displaystyle i\frac{\partial f}{\partial x}=i\left(\frac{\partial u}{\partial x}+i\,\frac{\partial v}{\partial x}\right)=$ $\displaystyle \left(\frac{\partial u}{\partial y}+i\,\frac{\partial v}{\partial y}\right)=\frac{\partial f}{\partial y}$
Now just compare real and imaginary parts in both sides.
Tonio