Finding Symmetric Poles for Complex Function Integrals

In summary, the conversation discusses the search for a function with two simple poles and symmetrical properties with respect to the real and imaginary axes. After considering functions such as f(z) = 1/(z^4) and f(z) = 1/(z^5), which do not have simple poles and have divergent integrals, the suggestion is made to locate the poles symmetrically along the line x=y. However, this does not work and the poles are instead located on the line y = -x, which results in the integral along the imaginary axis being the negative of the integral along the real axis. This successfully solves the problem.
  • #1
lonewolf5999
35
0
I'm looking for a function which has two simple poles, and whose integral along the positive real axis from 0 to infinity is equal to its integral along the positive imaginary axis.

I don't really know where to start. I'm looking at functions which have symmetry with respect to real/imaginary axes, i.e. if x is real, then f(x) = f(ix), which led me to consider things like f(z) = 1/(z^4) or f(z) = 1/(z^5), but those don't have simple poles and their integrals from 0 to infinity on the axes don't exist since they blow up at the origin. Any help is appreciated!
 
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  • #2
Try locating your poles symmetrically with respect to the two axes. Like along the line x=y.
 
  • #3
After trying your suggestion with z1 = 1 + i, z2 = -1 - i, and f(z) = 1/((z - z1)*(z-z2)), I ended up with the integral over the imaginary axis being the negative of that over the real axis, so I located my poles on the line y = -x instead, and that solved the problem.
Thanks for the help!
 

FAQ: Finding Symmetric Poles for Complex Function Integrals

What is the integral of a complex function?

The integral of a complex function is a mathematical operation that is used to find the area under the curve of a complex function. It is an extension of the integral concept in calculus that is used for real-valued functions.

How is the integral of a complex function evaluated?

The integral of a complex function is evaluated using techniques from complex analysis, such as Cauchy's integral formula, contour integration, and the residue theorem. These techniques involve manipulating the complex function and integrating it along a path in the complex plane.

What is the significance of the integral of a complex function?

The integral of a complex function has various applications in physics, engineering, and other fields. It can be used to solve problems involving electric fields, fluid flow, and quantum mechanics. It also plays a crucial role in the study of functions of a complex variable.

Can the integral of a complex function have complex values?

Yes, the integral of a complex function can have complex values. This is because the integrand (the function being integrated) and the limits of integration can both be complex numbers. In fact, the integral can have infinitely many complex values, depending on the path of integration chosen.

Are there any special properties of the integral of a complex function?

Yes, there are several special properties of the integral of a complex function. For example, if the function is analytic (meaning it can be represented by a power series), then its integral is also analytic. Additionally, the integral of a function over a closed path is equal to the sum of its residues inside the path, according to the Cauchy residue theorem.

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