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Thread: Integration

  1. #1 Integration 
    Forum Masters Degree thyristor's Avatar
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    Hi!
    Working on a problem involving Planck's radiation law, I am going to need to compute the integral , where T is the temperature.
    However, the integral cannot be solved analytically, so I need to solve it numerically, yet get an expression in T.
    Any ideas?


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  3. #2  
    Forum Ph.D. Heinsbergrelatz's Avatar
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    I guess im quite alright at solving integrals with exponential functions+ x etc.. Though i have no idea what planks radiation law even is, and what those big numbers divided by T represent, i can solve the integral in the question.

    In such questions like this you can assume a general case which is more convenient i think. Also these sort of integrals require some clever manipulations of the symbols.



    Now i see and , so i realized gamma function is the key here as the gamma function, of(n+1) is the integral of
    So basically all i had to do was rewrite this equation so we get e^{-x}x^{n} somehow.



    But before we get here we need to make a little substitution.

    So if we rewrite the whole thing now again: its like this (the sigma is nothing but the infinite series which is quite notable with the formula )



    the sum is with k=1 not zero because simply:



    Substitution:







    So just summing it up it becomes:





    I think this is how it goes...


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  4. #3  
    Forum Masters Degree thyristor's Avatar
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    Thanks a lot!
    The only problem is, your result is a sum which, correct me if I'm wrong, cannot be computed anaytically either
    I don't think that the result can be expressed in therms of elementary functions.
    However I made a plot in intervals of 100 K and got a pretty nice curve.
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  5. #4  
    Forum Ph.D. Heinsbergrelatz's Avatar
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    Thanks a lot!
    The only problem is, your result is a sum which, correct me if I'm wrong, cannot be computed anaytically either
    I don't think that the result can be expressed in therms of elementary functions.
    However I made a plot in intervals of 100 K and got a pretty nice curve.
    No probs. Im sure you know how to treat the gamma functions. Yes there was a sum, which to me seemed rather bizarre also, i was wondering so at school i asked my maths teacher, then he said that this sum you have here is the Riemann zeta function of . I dont know what riemann zeta function is, i have heard of it, bt i dont know how it works at my current level of understanding.
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  6. #5  
    Forum Masters Degree thyristor's Avatar
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    Sounds interesting. I have also just heard of the Riemann Zeta function, but I know it's related to prime numbers.

    For your interest, Planck's law of radiation gives the spectral radiance from a black body by the relationship:

    I wanted to find the efficiency for a light bulb by dividing the integral of over the visible spectrum, and divide it by the integral of over the entire spectrum, thus arriving at the above mentioned expression.
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  7. #6 Re: Integration 
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    Quote Originally Posted by thyristor
    Hi!
    Working on a problem involving Planck's radiation law, I am going to need to compute the integral , where T is the temperature.
    However, the integral cannot be solved analytically, so I need to solve it numerically, yet get an expression in T.
    Any ideas?
    we know the t is the temperature but can it be solved numerically with the rest of the puzzle
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