Blackbody Radiation

Seán M. Stewart recently joined Nazarbayev University in Astana where he is an associate professor of engineering mathematics. Before moving to Kazakhstan, he spent eleven years working at The Petroleum Institute in Abu Dhabi, United Arab Emirates, where he was an associate professor in the Department of Mathematics. His main research interests lie in the fields of applied mathematics and in the history of computation. R. Barry Johnson has been involved for over 40 years in infrared technology, lens design, optical systems design, and electro-optical systems engineering, and has used many of the devices and techniques described in this book, and knew many of the individuals mentioned therein. He developed the method of integrating Planck’s equation using the method of Gauss–Laguerre quadrature. Dr Johnson is a Senior Research Professor at Alabama A&M University and has been a faculty member at two other academic institutions engaged in optics education and research, employed by a number of companies, and has provided consulting services within the field. Dr. Johnson is an SPIE Fellow and Life Member, OSA Fellow, and is an SPIE past president (1987). He has been awarded many patents and has published numerous scientific and technical articles. Dr. Johnson was awarded the 2012 OSA/SPIE Joseph W. Goodman Book Writing Award for Lens Design Fundamentals, Second Edition. He is a perennial co-chair of the annual SPIE Conference Current Developments in Lens Design and Optical Engineering.

SECTION I: THE BLACKBODY PROBLEM



Chapter 1. Thermal radiation and the blackbody problem



1.1 Towards a solution to the blackbody problem



1.2 Planck and the blackbody problem



1.3 The work of the experimentalists



1.4 Thermal laws from dimensional analysis



1.5 Transition and new beginnings



SECTION II: THEORETICAL AND NUMERICAL MATTERS



Chapter 2. Theoretical developments



2.1 Spectral representations



2.2 Two important special functions



2.2.1 Polylogarithms



2.2.2 The Lambert W function



2.3 Two common spectral scales used to represent blackbody radiation



2.4 Other spectral scale representations



2.5 Ephemeral spectral peaks



2.6 Logarithmic spectral scales



2.7 The radiometric and actinometric cases



2.8 Normalized spectral exitance



2.9 The Stefan–Boltzmann law



2.9.1 The traditional approach



2.9.2 A polylogarithmic approach



2.10 Fractional functions of the first kind



2.11 Fractional functions of other kinds



2.12 Centroid and median wavelengths



2.13 The standard probability distribution and cumulative probability distribution functions for blackbody radiation



2.14 Infrared, visible, and ultraviolet components in the spectral distribution of blackbody radiation



Chapter 3. Computational and numerical developments



3.1 Approximations to the spectral exitance



3.1.1 The laws of Wien and Rayleigh–Jeans



3.1.2 Extended Wien and Rayleigh–Jeans approximations



3.1.3 Polynomial interpolation and logarithmic correction factors



3.1.4 Laurent polynomials and non-rational approximations of Erminy



3.2 Computation of the fractional function of the first kind



3.2.1 Series expansion methods



3.2.1.1 Large arguments



3.2.1.2 Small arguments



3.2.1.3 Division point