In-band Radiance: Integrating the Planck Equation Over a Finite Range

Above we analytically integrated the spectral radiance over the entire spectral range. The result, Eq. (20), is the well-known Stefan-Boltzmann law.  Similarly, Eq. (22) gives the integrated photon radiance. As useful as the Stefan-Boltzmann law is, for many applications a finite spectral range is needed.  To facilitate this, we compute the one-sided integral of the spectral radiance. We follow the method described by Widger and Woodall[2], using units of wavenumber.  Note that using other spectral units produces the same result, because it represents the same physical quantity.

Noting that ,  we get     .

The remaining integral can be integrated by parts[3]:

This gives

                            (23)

Testing shows that carrying the summation up to n = min(2+20/x, 512) provides convergence to at least 10 digits.   

Any finite range can be computed using two one-sided integrals:

Further, the complimentary integral is easily evaluated using (19):

A similar formula can be derived for the in-band photon radiance:

Again using ,  we get   .

Integrating by parts       so           

                               (24)

Equations (23) and (24) provide efficient formulas for computing in-band radiance.  Example C++ computer source code is provided in Appendix A.