# Integrating the Blackbody Curve, Part 2

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In Part 1, the series approximation for the integrated blackbody spectrum were

(2.1)

The first summation term dominates at large v. This term can be summed up into a closed-form expression as

$\begin{array}{ccc}\hfill \sum _{m=1}^{\mathrm{\infty }}\frac{{q}^{m}}{m}& =& \sum _{m=0}^{\mathrm{\infty }}\frac{{q}^{m+1}}{m+1}\hfill \\ \hfill & =& \sum _{m=0}^{\mathrm{\infty }}\underset{0}{\overset{q}{\int }}{x}^{m}\mathit{dx}\hfill \\ \hfill & =& \underset{0}{\overset{q}{\int }}\sum _{m=0}^{\mathrm{\infty }}{x}^{m}\mathit{dx}\hfill \\ \hfill & =& \underset{0}{\overset{q}{\int }}\frac{\mathit{dx}}{1-x}\hfill \\ \hfill & =& \underset{0}{\overset{1-q}{\int }}\frac{\mathit{dy}}{y},y=1-x\hfill \\ \hfill & =& -\mathrm{ln}y|{}_{y=1}^{1-q}\hfill \\ \hfill & =& -\mathrm{ln}\left(1-q\right)\text{.}\hfill \end{array}$

(2.2)

The formula for F(v) for large v now becomes

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

(2.3)

Applying this to the formula for large v gives an improvement of about half a magnitude, as shown in Figure 1.

For small v, the fit becomes somewhat worse. See Figure 2.

The improvement for small v must be offset by the infinite series already added in Part 1.

Unfortunately the remaining terms do not appear to have convenient closed-form summations. One approach would be to use the logarithmic series accelerations such as methods by Herbert H. H. Homeier. Another is to subtract off similar summations, pushing the error terms off to higher orders.

The story continues in Integrating the Blackbody Curve, Part 3.

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