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IntegrateSineCosineByReduction(AngouriMath.Entity,System.Int32,System.Int32)

Method (no overloads)

Summary

int sin(t)^p cos(t)^q dt for whole exponents of any sign, by the four standard
recurrences, ending on the nine integrands with both exponents in {-1, 0, 1}.

Remarks

This is the boundary the polynomial route stops at. The three substitutions in
IntegrateAPowerOfSineTimesAPowerOfCosine(AngouriMath.Entity,PeterO.Numbers.ERational,PeterO.Numbers.ERational,AngouriMath.Entity,AngouriMath.Entity) each leave a Laurent polynomial,
so they answer the cases where one exponent is odd and positive, or where both are even
and sum to at most -2. Everything else needs a logarithm somewhere, and
no rearrangement of a polynomial produces one: sin^2 cos^(-3) is
x^2/sqrt(1 + x^2) seen through the tangent substitution, and its antiderivative
holds an inverse hyperbolic sine.
The four recurrences, each of them one integration by parts written out:

q down   I(p,q) =  sin^(p+1) cos^(q-1)/(p+q)   + ((q-1)/(p+q))     I(p, q-2)
p down   I(p,q) = -sin^(p-1) cos^(q+1)/(p+q)   + ((p-1)/(p+q))     I(p-2, q)
q up     I(p,q) = -sin^(p+1) cos^(q+1)/(q+1)   + ((p+q+2)/(q+1))   I(p, q+2)
p up     I(p,q) =  sin^(p+1) cos^(q+1)/(p+1)   + ((p+q+2)/(p+1))   I(p+2, q)

Termination is |p| + |q|, which every branch below lowers by two until
both exponents are in {-1, 0, 1}. The downward pair divides by p + q and
the upward pair by q + 1 and p + 1; the ordering is chosen so that the
branch taken never has a zero divisor. Raising is tried first for exactly that reason —
q + 1 is zero only at q = -1, which is already in range, so an exponent
at most -2 can always be raised, where p + q can vanish at any size.
It is closed: the recursion is on two integers walking towards a fixed set, and
it asks the integrator nothing.
https://github.com/asc-community/AngouriMath/issues/718

AngouriMath

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AngouriMath

IntegrateSineCosineByReduction​(AngouriMath.​Entity,​System.​Int32,​System.​Int32)

IntegrateSineCosineByReduction(AngouriMath.Entity,System.Int32,System.Int32)