# Deforming Contour of complex function

• yaychemistry
In summary, the speaker is seeking a way to obtain F(x), the values of a complex function along the real axis, using information known about the function along a contour in the complex plane. They have attempted to use the Cauchy-Riemann equations and integration steps, but have encountered some instability in more complex cases. They are asking for suggestions on a better method for this task.
yaychemistry
Hi,
First, thank you for reading this.

I've got a complex function, F(z), which is assumed to be analytic, and I know it's values along a contour in the complex plane. Say, for simplicity, that contour is known parametrically as x(t) = cos(t), y(t) = sin(t), 0 < t< pi, thus I know F(x(t) + i*y(t)), and possibly dF/dt.

What I want to do is obtain F(x), that is what is F along the real axis?

What I have done so far is to assume that F obeys the Cauchy-Riemann equations, then if
F(x,y) = u(x,y) + i*v(x,y), I can get du/dy and dv/dy from the information in dF/dt (sets up a nice little matrix problem). Then I can take an integration step towards the real axis to get F at a new contour (e.g. y(t) = 0.99sin(t), x(t) = cos(t)), I can then find dF/dt along this contour (sadly, only through a finite difference formula, or interpolation) and then find du/dy and dv/dy again, take a new step, etc.. until y(t) = 0. It works pretty well for simple test cases, such as, F(z) = z^n, etc. However, it seems to be a little unstable as well when F gets a little more complicated, that is errors tend to build and sometimes the function "explodes" before I get to the real axis.

What I would like to know is if this has been done (in a better way) by someone else, and where I could go about looking. My google searches haven't resulted in much and I don't know my way around the math journals. Any suggestions?

Thanks!

This can be done. If F(z) is analytic and defined over some contour of nonzero length, it should be possible to analytically continue F to the whole complex plane (in a unique way).

The Kramers-Kronig relations demonstrate one example. If F is defined on either the real axis or the imaginary axis, the entire function can be found by doing some integrals.

## 1. What is a deforming contour of a complex function?

A deforming contour is a curve or path that is used to study the behavior of a complex function. It is typically a closed curve that is continuously deformed while keeping its endpoints fixed, allowing for a better understanding of the function's properties and behavior.

## 2. How is a deforming contour used in complex analysis?

In complex analysis, a deforming contour is used to evaluate integrals of complex functions. By continuously deforming the contour, it allows for simplification of the integral and can reveal important information about the function, such as its singularities or poles.

## 3. What is the relationship between a deforming contour and the Cauchy integral theorem?

The Cauchy integral theorem states that the integral of a complex function over a closed contour is equal to 0 if the function is analytic within the region enclosed by the contour. A deforming contour allows for the application of this theorem by simplifying the integral and showing that it is equal to 0.

## 4. Can a deforming contour be used for functions with multiple variable inputs?

Yes, a deforming contour can be used for functions with multiple variable inputs. In this case, the contour is a multidimensional path that is continuously deformed, allowing for the evaluation of multidimensional integrals.

## 5. What are some practical applications of using a deforming contour in scientific research?

A deforming contour has various practical applications in scientific research, particularly in physics and engineering. It can be used to solve complex integrals, determine the stability of physical systems, and study the behavior of electromagnetic fields, among others.

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