Complex Number Integral: Tips and Tricks for Solving

In summary, an integral with complex numbers is a mathematical concept that involves complex numbers in the limits of integration, integrand, or both. It follows the same rules as traditional integrals and is used in various fields such as signal processing, quantum mechanics, and electromagnetism. The convergence of an integral with complex numbers depends on the convergence of its real and imaginary parts, and it can have multiple solutions due to the different representations of complex numbers. It is important to specify the representation being used when solving such integrals.
  • #1
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[SOLVED] integral with complex numbers

Hi,

I feel really silly for asking this question... but can anyone give me any advice on how to evaluate the following intergral:

[tex]\int\int\int[/tex]r.e[tex]^{iqrcos(\theta)}[/tex]sin([tex]\theta)[/tex]dr.d([tex]\theta)}[/tex].d([tex]\phi)}[/tex]

basically an integral spanning whole of space (I've included the the jacobian r[tex]^{2}[/tex] and sin([tex]\theta)[/tex])

I've tried splitting it into cos(x)+isin(x) but it gets very, very messy:(

Another question:

Is there any way to do braket notation through latex?
 
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  • #2
nevermind... figured out how to solve it - simple substitution!
 

1. What is an integral with complex numbers?

An integral with complex numbers is a mathematical concept where the limits of integration, the integrand, or both contain complex numbers. It is a generalization of the traditional integral, which only deals with real numbers.

2. How is an integral with complex numbers evaluated?

The evaluation of an integral with complex numbers follows the same rules as traditional integrals. The complex numbers are treated as constants, and the integral is solved using integration techniques such as substitution, integration by parts, or partial fractions.

3. What are the applications of integrals with complex numbers?

Integrals with complex numbers are used in various fields of science and engineering, including signal processing, quantum mechanics, and electromagnetism. They are also used in solving differential equations and in the study of complex functions.

4. How do complex numbers affect the convergence of an integral?

The convergence of an integral with complex numbers is determined by the convergence of its real and imaginary parts. If both parts converge, the integral is said to be absolutely convergent. If only one part converges, the integral is conditionally convergent. If neither part converges, the integral diverges.

5. Can an integral with complex numbers have multiple solutions?

Yes, an integral with complex numbers can have multiple solutions. This is because complex numbers have multiple representations, such as polar form and rectangular form, which can lead to different solutions. It is important to specify which representation is being used when solving an integral with complex numbers.

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