Hi,
When I input
y = symbols("y", positive=True)
z = symbols("z", real=True)
K = symbols("K", integer=True, positive=True)
eta = Symbol("eta", positive=True)
g = y * ((1-(z*y)**2)**(-2+K/2 + eta) * gamma(S(1)/2 *
(-1+K)+eta))/(sqrt(pi) * gamma(-1+K/2+eta)) * (2 * (y**2)**eta
(1-y**2)**(S(1)/2 * (-4+K)))/beta(S(1)/2+eta,-1+K/2)
f = simplify(integrate(g, (y, 0, 1), meijerg=True))
f
I obtain
⎛ K │ ⎞
⎜ ─ - 2 │ ⎟
⎜ 2 K │ ⎟
2 ⎜η (0) + 1, - ─ - η + 2 │ ⎟
2 ⎛K 1⎞ ┌─ ⎜ 2 │ 2 2⋅ⅈ⋅π⎟
η (0)⋅Γ⎜─ + η - ─⎟ ⋅ ├─ ⎜ │ z ⋅ℯ ⎟
⎝2 2⎠ 2╵ 1 ⎜ K │ ⎟
⎜ ─ - 2 │ ⎟
⎜ 2 │ ⎟
⎝ η (0) + 2 │ ⎠
─────────────────────────────────────────────────────────────────
⎛ K ⎞
⎜ ─ ⎟
___ ⎜ 2 2 ⎟ ⎛K ⎞ ⎛K ⎞
╲╱ π ⋅⎝η (0) + η (0)⎠⋅Γ⎜─ - 1⎟⋅Γ(η + 1/2)⋅Γ⎜─ + η - 1⎟
⎝2 ⎠ ⎝2 ⎠
But I do not know how to interpret the eta(0) notation. In this problem,
eta is a fixed positive scalar, so I am not sure how an operator becomes
involved.
FWIW, Mathematica also produces an answer in terms of the Gaussian
hypergeometric function that is perhaps equivalent but without any unusual
notation involving eta. However, I had to supply Mathematica with the
assumption that z^2 < 1, and I am trying to understand the function, f, for
any real z. (although I'm pretty sure it will be a piecewise function). The
function should always output a real number, but Mathematica gives me a
complex result if I assume z^2 > 1.
Thanks,
Ben
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