Back to Contents

    Employing our favorite trick, we convert the integrand into a form that we have already integrated, thereby making the problem trivial. In this case we want to transform a fraction whose denominator consists of the product of two simple factors into the difference between two fractions whose denominators each comprise one of those factors; thus, we have

(Eq'n 1)

in which we must necessarily have

(Eq'n 2)

We know that m, n, a, and b represent constants, so we must have mbx-nx=0; thus we have n=mb. Then it follows trivially that m=1/a and we have equation 1 as

(Eq'n 3)

Our integral thus becomes

(Eq'n 4)

Example Application:

Spectral Density of Blackbody Radiation

    In October 1900 Max Planck first devised the law that bears his name, a mathematical description of the spectral density of the electromagnetic radiation emanating from a perfectly black body. In his Nobel lecture, 1920 Jun 02, he offered a verbal sketch of the process of reasoning that he had used to make the discovery, one different from the process of reasoning that he described in the paper he had published in Annalen der Physik in early 1901, one that did not use Ludwig Boltzmann's statistical thermodynamics.

    As the Nineteenth Century turned into the Twentieth physicists sought to gain a theoretical understanding of blackbody radiation by envisioning a cavity with perfectly black walls and imagining filling that cavity with an array of some kind of electromagnetic resonators. In the state in which the resonators have come into thermal equilibrium with the radiation the relationship between the entropy S of a single resonator and that resonator's average energy U will offer clues to the distribution of energy within the radiation.

    On October 07 of 1900 Planck already had in hand data indicating that in the high-frequency part of the electromagnetic spectrum blackbody radiation conforms to the relation

(Eq'n 5)

That afternoon, over tea, he got information that at low frequencies blackbody radiation conforms to the relation

(Eq'n 6)

Wanting a relationship that would cover the whole electromagnetic spectrum, Planck devised a compromise function and guessed that

(Eq'n 7)

He then integrated that formula with respect to energy so that he could exploit the thermodynamic law that

(Eq'n 8)

Thus Planck had

(Eq'n 9)

Finally he solved that equation for U and got

(Eq'n 10)

Later work would show that α=-k (k=Boltzmann's constant) and β=hν, the quantum of energy that the resonator absorbs or emits in its interaction with the radiation field. Thus we now have the Planck function

(Eq'n 11)

efefefaaabbbefefef

Back to Contents