Euler equation

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1. Jul 3, 2015

Galizius

when I am using Euler equation for Fourier transform integrals of type $$\int_{-\infty}^{\infty} dx f(x) exp[ikx]$$I am getting following integrals:

$$\int_{-\infty}^{\infty} dx f(x) cos(kx)$$ (for the real part) and

$$i* \int_{-\infty}^{\infty} dx f(x) sin(kx)$$ (for its imaginary part)

I am wondering what is the final integration result though. Is that the sum of both parts or are they seperate results? And if it is sum, when the imaginary or real part is being reduced to 0

Last edited: Jul 3, 2015
2. Jul 3, 2015

Dr. Courtney

The Fourier transform is the sum of both real and imaginary parts.

3. Jul 3, 2015

HallsofIvy

Staff Emeritus
Surely if you know that $e^{ikx}= cos(kx)+ i sin(kx)$ then you know that $\int f(x)e^{ikx}dx= \int (f(x)cos(kx)+ if(x)sin(kx))dx= \int f(x)cos(kx) dx+ i \int f(x)sin(kx) dx$.

4. Jul 3, 2015

Dr. Courtney

Well, when you put it that way ...

Nice proof. Thanks.

5. Jul 5, 2015

Awais ijaz

what is the complete form of euler equation?

6. Jul 5, 2015