Complex exponential (properties)

[gomez]

Hi, I am solving a second order ODE. the result I got is an exponential to the power of a real and an imaginary part, both of them inside a square root. I need to brake this result into an imaginary and a real part because in this particular case just the imaginary part of the solution is my solution. My question is How can a brake exp (square root of (4+i))?

thanks

gomez

 

[dextercioby]

Post the ODE.

Daniel.

 

[gomez]

The ODE is f(r)'' + 1/R* f'(r) - (i+1/r^2)f(r)=0 ; and my boundary conditions are F(r=r1)=1 and F(r=r2) =0. I solved this ODE and I found my two constants but this ODE comes from a PDE which boundary condition is sine(t) whis is the imaginary part of exp(it). my specific question is how can I break and exponential function that comes as a solution of the ODE? I need this solution to have two parts one imaginary and one real, because my solution will be just the imaginary part... is something similar to stoke's problem solution ( fluid mechanics )

 

[dextercioby]

The equation is quasi linear,i don't think it can be solved that easy.That nonconstant factor spoils everything.I think a numerical solution is the only answer.

Daniel.

 

[gomez]

Again, my specific question is about the exponential fuction, my solution is exp(root square(4+i)), how can I break this expression into a real part and an imaginary part. is there any property of the exponential function with complex numers that I'm missing?.

thanks

pd: the number 4 in my solution just indicates any real number.

 

[dextercioby]

$$(4+i)^{1/2}=...?$$

$$4+i=\sqrt{17}\left(\cos\arctan \frac{1}{4}+i\sin\arctan \frac{1}{4}\right)$$ (1)

Then

$$(4+i)^{1/2}=17^{1/4}\left(\cos\frac{\arctan\frac{1}{4}}{2}+i\sin\frac{\arctan\frac{1}{4}}{2}\right)$$ (2)

Simple stuff.

Daniel.

P.S.(as an edit) I hope u know how to exponentiate that animal (#2),don't u?

 

[gomez]

thanx man, you are really good at this.

P.S: I don't really have to take the exponential of that, I will just take the imaginary part and represent it as my solution, exp(imaginary).

P.S.2: people in my lab are still laughing about your "simple stuff" comment

thanks again

 

[dextercioby]

I don't know what u have to do,i said it's weird that u asked for such a simple thing,when the ODE u posted looks awfully difficult.

Daniel.