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The beta function
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addition theorem algorithm analytic arguments assume Bessel functions beta function binomial branch point Carlson Chapter choose coefficients compact subset complex con(z continuous converges uniformly convex Corollary cos2 defined definition denote differential equations Dirichlet average Dirichlet measure disk duplication theorem elementary ellipsoid elliptic functions elliptic integrals entire function Euler example Exercise finite formula Fubini's theorem functional relations gamma function Gauss's Gegenbauer polynomials half-plane Hence holomorphic homogeneity hypergeometric implies inequality integer integrand inverse Jacobi polynomials Laguerre polynomials left side Legendre polynomials Let n e line segment linear log-convex logarithmic Multiply obtain parameters permanence of functional permutation symmetry plane positive integer Proof prove quadratic transformation real axis real line replace representation right side Section series converges Show sin2 special functions standard functions Taylor series term by term variables Weierstrass zero