Calculating a complex integral

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SUMMARY

The discussion centers on calculating a complex integral that cannot utilize the residue theorem due to the absence of singularities. Participants suggest expressing the integral in terms of the Fourier transform of a Gaussian function, which provides a viable approach for evaluation. This method leverages established mathematical principles to simplify the computation of the integral.

PREREQUISITES
  • Understanding of complex analysis and integrals
  • Familiarity with the residue theorem
  • Knowledge of Fourier transforms, specifically Gaussian functions
  • Basic skills in mathematical proofs and transformations
NEXT STEPS
  • Research the properties of the Fourier transform of Gaussian functions
  • Study complex integrals and their evaluation techniques
  • Explore advanced applications of the residue theorem
  • Learn about alternative methods for integral evaluation in complex analysis
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Mathematicians, physics students, and anyone involved in advanced calculus or complex analysis who seeks to deepen their understanding of integral evaluation techniques.

Robin04
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Homework Statement
##f(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dt \ e^{-\frac{t^2}{2\sigma^2}+t(i\omega-\alpha)}cos(\Omega t)##
Relevant Equations
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As this function has no singularities the residue theorem cannot be applied. Can you help me a bit?
 
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Looks like you can express it in terms of the Fourier transform of a Gaussian.
 

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