Evalution of a complex integral

In summary, the conversation discusses the evaluation of the integral \displaystyle \int e^{-ix^{2}} \ dx without using a closed contour and the residue theorem. It is suggested to use the asymptotic expansion of the error function and take the limit as R goes to \infty. The reference provided discusses the Fresnel integrals and the integral from 0 to infinity.
  • #1
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Is there a problem with the following evaluation?[itex]\displaystyle \int e^{-ix^{2}} \ dx = \frac{1}{\sqrt{i}} \int e^{-u^{2}} \ du = \left( \frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \right) \text{erf}(u) + C = \left(\frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \right) \text{erf} (\sqrt{i}x) + C[/itex] So [itex] \displaystyle \int_{0}^{\infty} e^{-ix^{2}} \ dx = \left(\frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \right) \text{erf} (\sqrt{i}x) \Big|^{\infty}_{0} = \left(\frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \right) \text{erf} (\sqrt{i} \infty) [/itex]

or more precisely [itex] \displaystyle \left(\frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \right) \lim_{R \to \infty} \text{erf} (\sqrt{i} R) [/itex]The error function has an essential singularity at [itex] \infty [/itex] , so the limit as you approach [itex] \infty [/itex] is path dependent. But aren't we looking specifically for the limit as we approach [itex] \infty [/itex] on the line that originates at the origin and makes a 45 degree angle with the positive real axis?

So my idea was to use asymptotic expansion of the error function ([itex] \displaystyle 1 - e^{-x^{2}} O \left( \frac{1}{x} \right) [/itex]), replace [itex] x [/itex] with [itex] \sqrt{i} R[/itex], and take the limit as [itex]R[/itex] goes to [itex] \infty [/itex]. Is that valid?
 
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  • #3
mathman said:
http://en.wikipedia.org/wiki/Fresnel_integral

exp(-ix2) = cos(x2) - isin(x2).

Above reference discusses the integrals as well as the integral from 0 to infinity.
I want to evaluate the integral without using a closed contour and the residue theorem.
 
  • #4
You can carry out the integrals for the cos and sin from 0 to T and let T -> ∞.
 

Related to Evalution of a complex integral

1. What is the purpose of evaluating a complex integral?

The purpose of evaluating a complex integral is to determine the value of the integral over a complex plane, which is useful in many areas of mathematics and physics.

2. How is a complex integral different from a real integral?

A complex integral involves integrating a function over a complex plane, while a real integral involves integrating a function over a real line. This means that a complex integral takes into account both real and imaginary components of the function.

3. What methods are used to evaluate complex integrals?

There are several methods for evaluating complex integrals, including the Cauchy integral formula, the residue theorem, and contour integration. Each method is useful for different types of complex integrals.

4. What are some applications of evaluating complex integrals?

Evaluating complex integrals is useful in many areas of mathematics and physics, including in the study of complex functions, Fourier analysis, and solving differential equations. It also has practical applications in engineering and signal processing.

5. What are some challenges in evaluating complex integrals?

Evaluating complex integrals can be challenging because it involves working with complex numbers and complex functions, which can be difficult to visualize and manipulate. It also requires a deep understanding of complex analysis and various integration techniques.

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