# Complex exponential (properties)

• gomez
In summary, 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 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...
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

Post the ODE.

Daniel.

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 )

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.

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.

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

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

Last edited:
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

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.

## What is a complex exponential?

A complex exponential is a mathematical function of the form eix, where i is the imaginary unit and x is a complex number.

## What are the properties of a complex exponential?

Some properties of a complex exponential include:

• The value of eix is always a complex number with a magnitude of 1.
• The complex exponential follows the same rules of exponents as real numbers.
• The complex exponential is periodic, with a period of 2π.

## How is a complex exponential graphed?

A complex exponential is graphed on a complex plane, with the real part of the complex number on the x-axis and the imaginary part on the y-axis.

## What is the relationship between complex exponentials and trigonometric functions?

There is a close relationship between complex exponentials and trigonometric functions. Specifically, eix = cos(x) + i*sin(x). This relationship is known as Euler's formula.

## What are some applications of complex exponentials?

Complex exponentials are used in many fields, including signal processing, electrical engineering, and quantum mechanics. They are also used in solving differential equations and modeling periodic phenomena.

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