Inband Radiance: Integrating the Planck Equation Over a Finite Range
(click on equations to view enlarged)
Above we analytically integrated the spectral radiance over the entire spectral range. The result, Eq. (20), is the wellknown StefanBoltzmann law. Similarly, Eq. (22) gives the integrated photon radiance. As useful as the StefanBoltzmann law is, for many applications a finite spectral range is needed. To facilitate this, we compute the onesided 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 onesided integrals:


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


A similar formula can be derived for the inband photon radiance:
Again using , we get .
Integrating by parts so
(24)
Equations (23) and (24)
provide efficient formulas for computing inband radiance. Example C++ computer source code is provided
in Appendix A.
Calculation of a Blackbody Radiance 
Units of Frequency 
Units of Wavelength 
Units of Wavenumbers 
Radiance: Integrating the Planck Equation 
Inband Radiance: Integrating the Planck Equation over a Finite Range 
Appendix A: Algorithms for Computing Inband Radiance 
Appendix B: The Doppler Effect 
Appendix C: Summary of Formulas 
References 
Blackbody Calculator 
Print Version 
[2] Widger, W. K. and Woodall, M. P., Integration of the Planck blackbody radiation function, Bulletin of the Am. Meteorological Society, 57, 10, 12171219, Oct. 1976
[3] CRC Handbook of Chemisry and Physics, 56^{th} edition #521