The Planck Function Expressed as B_ν(T) and B_λ(T).

We begin by expressing the Planck function

B_λ(T) = (2hc²/λ⁵)/(e ^hc/λkT - 1).

where, in cgs units, [B_λ(T)] = ergs s^-1 cm^-2 sterad^-1 cm^-1 = ergs s^-1 cm^-3 sterad^-1.  We would also like to express the radiant power per unit frequency interval, i.e., [B_ν(T)] = ergs s^-1 cm^-2 sterad^-1 Hz^-1= ergs cm^-2 sterad^-1.

Because of energy conservation, it must be that B_ν(T) dν = -B_λ(T) dλ, and, since λν = c, it also follows that
dν = -cλ^-2 dλ.  A little algebraic manipulation then shows that the Planck function can also be expressed

B_ν(T) = (2hν³/c²)/(e ^hν/kT - 1).

(Note:  Since B_λ(T) ¹ B_ν(T), these two functions do not peak at the same wavelength, nor at the same frequency.)