# Some complex Fourier-like infinite integral

• Ahmes
In summary, the conversation discusses the given integral and its solution, which involves the use of the known formula for \int_{-\infty}^{\infty} e^{ikx} dx and a further derivation of the real and imaginary parts. The explanation for the imaginary part is contested and deemed incorrect by the person asking for help.

## Homework Statement

The following integral is given:
$$\int_{0}^{\infty} e^{ikx} dx$$
k & x are real.

## Homework Equations

We know that:
$$\int_{-\infty}^{\infty} e^{ikx} dx=2\pi \delta(k)$$

## The Attempt at a Solution

$$\mathcal{I}=\pi \delta(k) + i \frac{1}{k}$$
I could easily prove the real part with the formula in 2, but couldn't be persuaded why the imaginary part is so (it barely makes sense). I tried:
$$\Im \int_{0}^{\infty} e^{ikx} dx = \int_{0}^{\infty} =\sin(kx) dx = \lim_{M\rightarrow \infty} \frac{1}{k}(-\cos(kM)+1)$$
But $\lim_{M\rightarrow \infty} \cos(kM)$ is quite meaningless and I can't see why the whole expression should be k^{-1}...

I'll appreciate any help,
Thanks!

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## What is a complex Fourier-like infinite integral?

A complex Fourier-like infinite integral is a mathematical concept that involves integrating a complex-valued function over an infinite range. It is similar to a traditional Fourier integral, but instead of using real-valued functions, complex-valued functions are used.

## What is the purpose of a complex Fourier-like infinite integral?

The purpose of a complex Fourier-like infinite integral is to analyze complex-valued functions and their properties. It is commonly used in physics, engineering, and other fields to solve problems involving waves, signals, and other phenomena.

## How is a complex Fourier-like infinite integral different from a traditional Fourier integral?

The main difference between a complex Fourier-like infinite integral and a traditional Fourier integral is the use of complex-valued functions. This allows for a more comprehensive analysis of certain types of functions, such as those with non-zero imaginary components.

## What are some applications of complex Fourier-like infinite integrals?

Complex Fourier-like infinite integrals have numerous applications in various fields such as signal processing, image processing, and quantum mechanics. They are also used in solving differential equations and analyzing periodic functions.

## What are some techniques for solving complex Fourier-like infinite integrals?

There are several techniques for solving complex Fourier-like infinite integrals, including the use of contour integration, residue theorem, and Fourier transform. These techniques involve manipulating the complex-valued function and finding suitable paths of integration to simplify the integral.