1. 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?

2.

3. 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...

4. 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.

5. 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.

6. 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.

7. 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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