- #1
DianaSagita
- 11
- 0
1. This is something from complex analysis: Find the analytic function f(z)= f(x+iy) such that arg f(z)= xy.
2. [tex]w=f(z)=f(x+iy)=u(x,y)+iv(x,y) (*), w=\rho e^{i\theta} (**)[/tex]
Here are the Cauchy-Riemann conditions...
[tex]\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\,\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}[/tex]
And polar Cauchy-Riemann conditions...
[tex]\rho\frac{\partial u}{\partial \rho}=\frac{\partial v}{\partial \theta},\,\rho\frac{\partial v}{\partial \rho}=-\frac{\partial u}{\partial \theta}[/tex]
Also what might be helpful - Re and Im part of analytic function are harmonic functions, so:
[tex]\frac{\partial^2u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0[/tex]
(Laplace's differential equation)
3. Next what I tried to do (if it's correct)...
I transformed (*) to (**) so,
[tex]w=\sqrt{u^2+v^2} e^{i atan\frac{v}{u}}[/tex]
[tex]atan\frac{v}{u}=xy[/tex]
[tex]v=u\,tan(xy)[/tex]
[tex]w=u\sqrt{1+tan^2(xy)} e^{i xy}= \frac{u(x,y)}{cos(xy)} e^{i xy}=u(x,y)+i[u(x,y) tan(xy)][/tex]
[tex]{u_x^'}={u_y^'}tan(xy)+\frac{x u}{cos^2(xy)}[/tex]
Now, I assume I got to get system of partial differential equations (by using second condition)...
Is this ok, and is there any easier way of getting analytical function?
Main thing that confuses me is the given argument which is included in real and imaginary part...
Pls help!
2. [tex]w=f(z)=f(x+iy)=u(x,y)+iv(x,y) (*), w=\rho e^{i\theta} (**)[/tex]
Here are the Cauchy-Riemann conditions...
[tex]\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\,\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}[/tex]
And polar Cauchy-Riemann conditions...
[tex]\rho\frac{\partial u}{\partial \rho}=\frac{\partial v}{\partial \theta},\,\rho\frac{\partial v}{\partial \rho}=-\frac{\partial u}{\partial \theta}[/tex]
Also what might be helpful - Re and Im part of analytic function are harmonic functions, so:
[tex]\frac{\partial^2u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0[/tex]
(Laplace's differential equation)
3. Next what I tried to do (if it's correct)...
I transformed (*) to (**) so,
[tex]w=\sqrt{u^2+v^2} e^{i atan\frac{v}{u}}[/tex]
[tex]atan\frac{v}{u}=xy[/tex]
[tex]v=u\,tan(xy)[/tex]
[tex]w=u\sqrt{1+tan^2(xy)} e^{i xy}= \frac{u(x,y)}{cos(xy)} e^{i xy}=u(x,y)+i[u(x,y) tan(xy)][/tex]
[tex]{u_x^'}={u_y^'}tan(xy)+\frac{x u}{cos^2(xy)}[/tex]
Now, I assume I got to get system of partial differential equations (by using second condition)...
Is this ok, and is there any easier way of getting analytical function?
Main thing that confuses me is the given argument which is included in real and imaginary part...
Pls help!