Derivative of a complex function in terms of real and imaginary parts.

[gauss mouse]

Hi, I wonder if anyone knows when (maybe always?) it is true that, where

[itex]z=x+iy \text{ and } f : \mathbb{C} \to \mathbb{C} \text{ is expressed as } f=u+iv, \text{ that }
f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}?
[/itex]
I'm pretty sure that this is true for [itex]f=exp.[/itex]

I should be able to find this but searching google for mathematics is a nightmare.

 

[quZz]

Hi,
if [itex]f(z)[/itex] is analytical function, you can take derivative in any direction on complex plane of [itex]z[/itex], e.g. take it along real axis [itex]dz = dx[/itex].

 

[micromass]

Hi, I wonder if anyone knows when (maybe always?) it is true that, where

[itex]z=x+iy \text{ and } f : \mathbb{C} \to \mathbb{C} \text{ is expressed as } f=u+iv, \text{ that }
f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}?
[/itex]
I'm pretty sure that this is true for [itex]f=exp.[/itex]

I should be able to find this but searching google for mathematics is a nightmare.

This is indeed true. But remember that a complex differentiable function must satisfy Cauchy-Riemann equations!

I suggest looking at Theorem I.5.3 of "Complex Analysis" by Freitag and Busam. A free preview is available at google books.

 

[gauss mouse]

Hi,
if [itex]f(z)[/itex] is analytical function, you can take derivative in any direction on complex plane of [itex]z[/itex], e.g. take it along real axis [itex]dz = dx[/itex].

Thanks for your help. I should have specified that f be analytic.Say I wanted to take the derivative in another direction, not along the real or imaginary axis, how would I do that?

 

[gauss mouse]

This is indeed true. But remember that a complex differentiable function must satisfy Cauchy-Riemann equations!

I suggest looking at Theorem I.5.3 of "Complex Analysis" by Freitag and Busam. A free preview is available at google books.

Thanks for your help. That looks from the preview like a nice book. I'll see if I can find it in the college library.

 

[micromass]

Say I wanted to take the derivative in another direction, not along the real or imaginary axis, how would I do that?

This isn't really a question about real analysis, but rather about multivariable limits.

Say we have the unit vector [itex](a,b)[/itex]. To find the derivative in that direction, we can do this by

[tex]a\frac{\partial f}{\partial x}+b\frac{\partial f}{\partial y}[/tex]

This can be done if the function is differentiable at the point.

 

[quZz]

Well, you'll get the same answer because f(z) is analytic, it follows from definition. You can use Cauchy-Riemann equations to get different forms of df/dz.

 

[lavinia]

Hi, I wonder if anyone knows when (maybe always?) it is true that, where

[itex]z=x+iy \text{ and } f : \mathbb{C} \to \mathbb{C} \text{ is expressed as } f=u+iv, \text{ that }
f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}?
[/itex]
I'm pretty sure that this is true for [itex]f=exp.[/itex]

I should be able to find this but searching google for mathematics is a nightmare.

A complex differentiable function may be thought of as a real differentiable function whose Jacobian is multiplication by a complex number. As a linear transformation of R^2 a complex number is a rotation followed by a dilation. This gives you the relations between the partial derivatives of f.