Planck's law for spectral radiance of a black body for a wavelength:

$$P=\frac{2hc^2}{\lambda^5}\frac{1}{e^{hc/\lambda k_bT}-1}$$

If I want to know the spectral radiance for an interval of wavelengths, for example from $\lambda_1=0.5 \,\mu m$ to $\lambda_2=0.55\,\mu m$, I should integrate Planck's law:

$$P=\int_{\lambda_1}^{\lambda_2}\frac{2hc^2}{\lambda^5}\frac{1}{e^{hc/\lambda k_bT}-1}d\lambda$$

My question is how to take this integral? Probably somebody already have solved this, but I cannot find, any links would be appreciated

  • $\begingroup$ It doesn’t make sense to use $P$ to denote both quantities. They don’t even have the same dimensions. $\endgroup$
    – G. Smith
    Nov 28, 2020 at 18:35

1 Answer 1


how to take this integral?

This definite integral can be done analytically when you are integrating over all wavelengths from zero to infinity, but not when you are integrating over a finite interval like $\lambda_1$ to $\lambda_2$.

You can do it numerically instead. Since you have a small interval, the integrand doesn't change much over it. You could approximate the integral as the value of the integrand in the middle of the interval times the interval, approximating the area under the curve by a single rectangle. Or you can use a more sophisticated numerical integration scheme.


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