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 Integrating this results in

$ \int\limits_{0}^{\infin} F(\lambda) e^{-\frac c{\vartheta\lambda}} d\lambda = \frac c\vartheta \int\limits_0^{\infin} 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^\infin e^{-y} y^{n-2} dy $.|undefined

It therefore follows that

$ \mathrm{const.} \vartheta^4 = \sum_n a_n \frac{\vartheta^{n-1}}{c^{n-1}} \Gamma(n-1) $|undefined

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

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

The equation for $$\varphi_\lambda$$ now becomes

$ \varphi_\lambda = \frac C{\lambda^5}e^{-\frac c{\lambda\vartheta}} $.|undefined

From this it follows that

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

for

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

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

$$\frac{d^2\varphi}{d\lambda^2}$$ 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 $$ \lambda = \infin $$, then the curve is an asymptote to the $$\lambda$$-axis.