Some complex Fourier-like infinite integral

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.
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Homework Statement


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

Homework Equations


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

The Attempt at a Solution


The answer is:
[tex]\mathcal{I}=\pi \delta(k) + i \frac{1}{k}[/tex]
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:
[tex]\Im \int_{0}^{\infty} e^{ikx} dx = \int_{0}^{\infty} =\sin(kx) dx = \lim_{M\rightarrow \infty} \frac{1}{k}(-\cos(kM)+1)[/tex]
But [itex]\lim_{M\rightarrow \infty} \cos(kM)[/itex] 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.

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