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Bateman Manuscript Project Bessel Functions cn u dn cn(u coefficient Complete Elliptic integral confluent hypergeometric function Consider the integral d2 y dy defined Derive differential equation dn u sn dn(u dn² x dx e-ikx e-u² e-x² eivo equation 12 exercise expansion F₁ Fourier series Fourier transform Fourier-Bessel series Functions CHAPTER Gamma function Hence show Hypergeometric Equation Iv+1 Jacobian Elliptic Functions Jv+1 k₁ k² sn² k² t² k² z² Laplace transform Legendre polynomials Normal Elliptic Integral Orthogonality Property periodic with period qn² recurrence relations satisfies the differential Show that sn sin² sn(u sn² u sn² SPECIAL FUNCTIONS spherical harmonic substitution theorem Theta double series Theta functions variable Verify Σ Σ Υπ ძვ