# Integrating the Blackbody Curve, Part 3

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In Part 1 and Part 2, the series approximation for the integrated blackbody spectrum was

(3.1)

Having two separate formulas is awkward in practice. Combining the two requires a function that differentiates between small and large v, going to one for small v and to zero for large v. This type of function is a variation of the Sigmoid function. After trying various functions, the first form that worked okay was e-v2. After applying this function to both sides of equation 1.6 from Part 1, the result is

$F\left(v\right)={e}^{-{v}^{2}}+\frac{15}{{\mathrm{\pi }}^{4}}\sum _{m=1}^{\mathrm{\infty }}{e}^{-\mathit{mv}}\left[\frac{{v}^{3}}{{m}^{}}+\frac{3{v}^{2}}{{m}^{2}}+\frac{6v}{{m}^{3}}\right]+\frac{6\left({e}^{-\mathit{mv}}-{e}^{-{v}^{2}}\right)}{{m}^{4}}$

(3.2)

A plot of the first 5 partial sums is shown in Figure 1.

The new fit clearly works well for large and small v. For v near 1, the fit appears to converge slowly or not at all. The steep, downward spikes are places where the fit crosses the true function, so the error is zero.

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