## Planck blackbody summation

Posted by peeterjoot on November 21, 2012

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# Motivation

Here’s a silly exercise. I’m so used to seeing imaginaries in expressions, when I looked at the famous blackbody summation for an exponentially decreasing probability distribution

I imagined (sic) an imaginary in the exponential and thought “how can that converge?”. I thought things must somehow magically work out if the limits are taken carefully, so I derived the finite summation expressions using the old tricks.

# Guts

If we want to sum a discrete power series, say

we have only to take the difference

so we have, regardless of the magnitude of

Observe that the derivative of is

but we also have

We expect this and 1.2.6 to differ only by a constant. For 1.2.6, or , we have at the origin, the same as 1.2.8. Our conclusion is

a result that applies, no matter the magnitude of . Now we can form the Planck summation up to some discrete summation point (say )

I got this far and noticed there’s still an issue with . Taking a second look, I see that we have a plain old real exponential, something perhaps like \cref{fig:negativeExponentialPlot:negativeExponentialPlotFig1}.

\imageFigure{negativeExponentialPlotFig1}{Plot of }{fig:negativeExponentialPlot:negativeExponentialPlotFig1}{0.3}

It doesn’t really matter what the value of is, it will be greater than zero, so that we have for our sum

which is the Planck result.

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