Page:Über die Energievertheilung im Emissionspectrum eines schwarzen Körpers.pdf/6

Integrating this results in

$\int \limits _{0}^{\infty }F(\lambda )e^{-{\frac {c}{\vartheta \lambda }}}d\lambda ={\frac {c}{\vartheta }}\int \limits _{0}^{\infty }F\left({\frac {c}{y\vartheta }}\right)e^{-y}{\frac {dy}{y^{2}}}=\sum _{n}a_{n}{\frac {\vartheta ^{n-1}}{c^{n-1}}}\int \limits _{0}^{\infty }e^{-y}y^{n-2}dy$.

It therefore follows that

$\mathrm {const.} \vartheta ^{4}=\sum _{n}a_{n}{\frac {\vartheta ^{n-1}}{c^{n-1}}}\Gamma (n-1)$

So all but one of the coefficients will vanish, and the result will be

$\vartheta ^{n-1}=\vartheta ^{4}$,

And therefore $n=5$

From this, it therefore follows that

$F(\lambda )={\frac {\mathrm {const.} }{\lambda ^{5}}}$.

The equation for $\varphi _{\lambda }$ now becomes

$\varphi _{\lambda }={\frac {C}{\lambda ^{5}}}e^{-{\frac {c}{\lambda \vartheta }}}$.

From this it follows that

${\begin{aligned}{\frac {d\varphi }{d\lambda }}&=-{\frac {Ce^{-{\frac {c}{\lambda \vartheta }}}}{\lambda ^{6}}}\left(5-{\frac {c}{\lambda \vartheta }}\right),\\{\frac {d^{2}\varphi }{d\lambda ^{2}}}&={\frac {Ce^{-{\frac {c}{\lambda \vartheta }}}}{\lambda ^{7}}}\left(30-{\frac {12c}{\lambda \vartheta }}+{\frac {c^{2}}{\lambda ^{2}\vartheta ^{2}}}\right);\end{aligned}}$

for

$\lambda ={\frac {c}{5\vartheta }}$ we have ${\frac {d\varphi }{d\lambda }}=0$,

${\frac {d^{2}\varphi }{d\lambda ^{2}}}=-{\frac {5Ce^{-5}}{\lambda ^{7}}}$;

${\frac {d^{2}\varphi }{d\lambda ^{2}}}$ is negative, so the value corresponds to a maximum. We shall refer to this value as $\lambda _{m}$. The associated value of $\varphi$ is

$\varphi _{m}={\frac {C}{\lambda _{m}^{5}}}e^{-5}$.

Since both $\varphi$ and ${\frac {d\varphi }{d\lambda }}$ vanish for $\lambda =\infty$, then the curve is an asymptote to the $\lambda$-axis.