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Thread: Trigonometric function and Gaussian function integration

  1. #1 Trigonometric function and Gaussian function integration 
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    How to solve combination of Trigonometric function and Gaussian function integral? such as

    http://i.imgur.com/hyvgHoY.gif
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    yeah, I know Jilan. but quantum mechanics involves lots of such integrals.
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    It isn't that messy, it turns out. There is a neat trick, but the justification of it requires some advanced concepts like contour integrals in the complex plane. I will just give an overview.

    Let's denote the original integral by I, i.e.


    Step 1. Take advantage of the evenness of the integrand and Euler's formula to rewrite the integral like so:


    Step 2. Note that (this is just completing the square), so that we can rewrite the integral as


    Step 3. Make the substitution . The integral is now over a contour in the complex plane going from to , but I am going to state without justification for now that I can simply drop the imaginary parts from these limits (this is just an overview!), resulting in an ordinary Gaussian integral along the real axis:

    Last edited by btr; 06-09-2014 at 11:52 PM.
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  5. #5  
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    Quote Originally Posted by NewtonApple
    yeah, I know Jilan. but quantum mechanics involves lots of such integrals.
    I recommend picking up something with a table of integrals such as the CRC Standard Mathematical Tables and Formulae – 30th Edition. I suggest this because when doing quantum mechanics your job is to learn the theory and not learn math. The math is something you can do separately. Doing both simultaneously is less efficient. However, this is certainly not to say that you shouldn’t learn how to do the integrals. For that you should review your basic calculus text, pick up a text on advanced calculus as well as a text on mathematical physics. Here are some of the most well-known texts in mathematical physics;

    Mathematical Physics in the Physical Sciences – Third Edition by Mary Boas

    Mathematical Methods of Physics – Second Edition by Mathews and Walker

    Mathematical Physics by Eugene Butkov

    For some strange reason the text by Butkov can be very expensive though. However you can download all of these from the internet.
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