Solving Complex Number Equations with e^{\frac{1}{2} i n x}

[JasonRox]

How does...

$$e^{\frac{1}{2} i n x} = \sin{ \frac{1}{2} n x}$$

...where n is any positive integer and x is any angle.

I know about de Moivre's Theorem, but that can't be deduced from there.

There is also brackets around it, with some sort of greek letter on the outside. Looks like a vertheta or something. I know this doesn't help much, but I just wanted you to know. The sign is not present after the equal sign.

It would make sense if the sign (greek letter) is used to get rid of imaginary numbers somehow. Also, x would have to be one radian measure so that cos would always be eliminated. Because cos(pi n 1/2) will always be zero.

Maybe it doesn't even use de Moivre's Theorem.

I'm clueless.

Maybe the greek letter represents 1/i and therefore gets rid of the i.

Honestly, it doesn't even say what x is, it is actually that circle with the line across so that should represent any angle.

Can someone help me out here?

 

[matt grime]

It doesn't equal that

I think the sign you can't translat is the Im( ) thing in some fancy germanic script.

 

[arildno]

As it stands, it is completely incorrect, UNLESS the sign prior signifies the IMAGINARY PART of the complex number.
(Remember that a complex number can be described in terms of two REAL numbers, the real part, and the imaginary part of that number).

That's the only clue I can give you..

 

[matt grime]

let's see if the tex here has it:

$$\mathfrak{I}$$
or

$$\Im$$

 

[James R]

$$e^{inx} = \cos(nx) + i \sin(nx)$$

So, it is true that

$$\mbox{Im}(e^{inx}) = \sin (nx)$$

 

[JasonRox]

That was of no help at all. You told me what I already know.

I appreciate your response though.

I was just hoping that maybe I wasn't seeing something. I'll ask my prof, and I hope he's willing to help.

 

[JasonRox]

It doesn't equal that

I think the sign you can't translat is the Im( ) thing in some fancy germanic script.

I believe that might be it.

I have never seen that notation and the book never mentionned anything about it.

 

[JasonRox]

I got it now.

The fancy thing is an I, which you know what it means.

Makes sense a lot of sense now.

Thanks.

Note: It means imaginary part for the readers who are interested in knowing.