How Does the Integral of exp(-x^2*k+i*m*x) Behave as m Increases?

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SUMMARY

The integral of the function exp(-x^2*k + i*m*x) from 0 to 5 behaves according to the Riemann-Lebesgue lemma, indicating that as the parameter m increases, the integral approaches zero. The integrand can be expressed as a combination of complex exponentials, specifically 1/2(e^(mix - x^2/a^2) + e^(-imx - x^2/a^2)). The primitive function of exp(-x^2) is not elementary and is represented by the complex error function, which becomes difficult to define as m grows large. This behavior highlights the challenges in numerical approximation of the error function, particularly for large m values.

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  • Understanding of complex analysis, particularly complex exponentials.
  • Familiarity with the Riemann-Lebesgue lemma and its implications.
  • Knowledge of error functions and their numerical approximations.
  • Basic calculus skills, especially integration techniques involving limits.
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  • Learn about the properties and applications of the complex error function erfi.
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\int ^{5}_{0} \left cos(m*x)e^{- \frac{x^{2}}{a^{2}}} \right dx
 
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Well, you can first the integrand as:
\frac{1}{2}(e^{mix-\frac{x^{2}}{a^{2}}}+e^{-imx-\frac{x^{2}}{a^{2}}}), i=\sqrt{-1}
Complete the squares in the exponents to proceed a bit further.
 
arildno said:
Well, you can first the integrand as:
\frac{1}{2}(e^{mix-\frac{x^{2}}{a^{2}}}+e^{-imx-\frac{x^{2}}{a^{2}}}), i=\sqrt{-1}
Complete the squares in the exponents to proceed a bit further.


how to integrate the first part only i.e. exp(-x^2*k+i*m*x) for some defined limits, say 0 to b. Mathcad goves solution in the form of complex error function that is difficult to define as erfi (m) when m is large can't be defined (infinite)..but the function has zero values at higher m...
 
Last edited:
ersanjaynp said:
how to integrate the first part only i.e. exp(-x^2*k+i*m*x) for some defined limits, say 0 to b. Mathcad goves solution in the form of complex error function that is difficult to define as erfi (m) when m is large can't be defined (infinite)..but the function has zero values at higher m...

The primitive function of exp{-x^2} is not elementary, therefore cannot be expressed in simpler terms than the error function which is numerically approximated.

As for the values of the integral as m tends to large numbers, this is an instance of the Riemann-lebesgue lemma, which generally says that for any smooth function f(t) the integral \int_{interval}cos(wt)f(t)dt will tend to zero as w tends to infinity
 

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