The integral for fo2
can be computed exactly, which makes the numerical computation of rms
straightforward:
Clear[s, gaussian2, fo2];
sigmaS = 50*10^-6;
gaussian2[s_] := 1/(Sqrt[2 Pi] sigmaS) Exp[-(s^2/(2 sigmaS^2))];
fo2[s_] = Integrate[gaussian2'[sprime]/(s - sprime)^(1/3), {sprime, -Infinity, s},
Assumptions -> s < 0];
rms2 = Sqrt[
NIntegrate[gaussian2[x] (fo2[x])^2, {x, -Infinity, Infinity}] -
(NIntegrate[gaussian2[x] fo2[x], {x, -Infinity, Infinity}])^2
]
(*
77003.5
*)
Note: The integral for fo2
evaluates to a ConditionalExpression
with Re[s] > 0
, if no assumptions are given or if the assumption that s
is real is given. However we get two results with either assumption that s > 0
or s < 0
, both of which are equivalent. Hence the assumption s < 0
above.
intpos = Integrate[
gaussian2'[sprime]/(s - sprime)^(1/3), {sprime, -Infinity, s},
Assumptions -> s > 0]
(*
180000/7 10^(1/3) ((7 Gamma[2/3] Hypergeometric1F1[2/3, 1/2, -200000000 s^2])/
Gamma[-(5/3)] - (
7500 2^(1/6)
s Gamma[13/3] Hypergeometric1F1[7/6, 3/2, -200000000 s^2])/
Sqrt[π])
*)
intneg = Integrate[
gaussian2'[sprime]/(s - sprime)^(1/3), {sprime, -Infinity, s},
Assumptions -> s < 0]
(*
(1/Sqrt[π])200000 5^(1/3) (Gamma[5/6] Hypergeometric1F1[2/3, 1/2, -200000000 s^2] -
10000 Sqrt[2]
s Gamma[4/3] Hypergeometric1F1[7/6, 3/2, -200000000 s^2])
*)
intpos - intneg // FullSimplify
(*
0
*)