Integrating the Blackbody Curve, Part 3


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In Part 1 and Part 2, the series approximation for the integrated blackbody spectrum was

F ( v ) = 15 π 4 ∑ m = 1 ∞ e − mv [ 3 v 2 m 2 + 6 v m 3 + 6 m 4 ] − 15 π 4 v 3 ln ( 1 − e − v ) for large  v = 1 + 15 π 4 ∑ m = 1 ∞ e − mv [ v 3 m + 3 v 2 m 2 + 6 v m 3 + 6 1 − e mv m 4 ] for small  v . matrix { alignr F(v) # {}={} # alignl 15 over { %pi^4 } SUM from m=1 to infinity { e^-mv left [ { 3 v^2 } over {m^2} + {6 v} over {m^3} + 6 over {m^4} right ] } - 15 over { %pi^4 } v^3 ln( 1 - e^{-v} ) # "for large " v ## {} # {} # {} # {} ## alignr {} # {}={} # alignl 1 + 15 over { %pi^4 } SUM from m=1 to infinity { e^-mv left [ v^3 over {m^{}} + {3 v^2 } over {m^2} + {6 v} over {m^3} + 6 {1 - e^mv} over {m^4} right ] } # "for small " v "." }

(3.1)

Having two separate formulas is awkward in practice. Combining the two requires a function that differentiates between small and large v, going to one for small v and to zero for large v. This type of function is a variation of the Sigmoid function. After trying various functions, the first form that worked okay was e-v2. After applying this function to both sides of equation 1.6 from Part 1, the result is

F ( v ) = e − v 2 + 15 π 4 ∑ m = 1 ∞ e − mv [ v 3 m + 3 v 2 m 2 + 6 v m 3 ] + 6 ( e − mv − e − v 2 ) m 4 F(v) = e^{-v^2} + 15 over { %pi^4 } SUM from m=1 to infinity { e^-mv left [ v^3 over {m^{}} + {3 v^2 } over {m^2} + {6 v} over {m^3} right ] + { 6 left ( e^-mv - e^{-v^2} right ) } over {m^4} }

(3.2)

A plot of the first 5 partial sums is shown in Figure 1.

Figure 1: Errors in series approximation to Blackbody curve for large and small v

The new fit clearly works well for large and small v. For v near 1, the fit appears to converge slowly or not at all. The steep, downward spikes are places where the fit crosses the true function, so the error is zero.


Categories Physics, Optics

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