ToDistribution
ToDistribution[exp, x]
replaces (1-x)^(a Epsilon - 1)
in exp
by 1/(a Epsilon) DeltaFunction[1-x] + 1/(1-x) + a Epsilon Log[1-x]/(1-x) + 1/2 a^2 Epsilon^2 Log[1-x]^2/(1-x)]
and (1-x)^(a Epsilon - 2)
in exp
by -1/(a Epsilon) DeltaFunctionPrime[1-x] + 1/(1-x)^2 + (a Epsilon) Log[1-x]/(1-x)^2 + a^2 Epsilon^2/2 Log[1-x]^2/(1-x)^2 + a^3 Epsilon^3/6 Log[1-x]^3/(1-x)^2
.
See also
Overview, PlusDistribution.
Examples
ToDistribution[(1 - x)^(Epsilon - 1), x, PlusDistribution -> pd]
61ε3pd(1−xlog3(1−x))+21ε2pd(1−xlog2(1−x))+εpd(1−xlog(1−x))+pd(1−x1)+εδ(1−x)
ToDistribution[(1 - x)^(Epsilon - 2), x, PlusDistribution -> Identity]
−εδ′(1−x)+6(1−x)2ε3log3(1−x)+2(1−x)2ε2log2(1−x)+(1−x)2εlog(1−x)+(1−x)21
Series2[Integrate[(1 - x)^(Epsilon - 2), {x, 0, 1}, GenerateConditions -> False], Epsilon, 3]
−ε3−ε2−ε−1
Integrate2[ToDistribution[(1 - x)^(Epsilon - 2), x], {x, 0, 1}]
−ε3−ε2−ε−1