## A treatise on special functions, for scientists and engineers |

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### Contents

SOLUTION OF DIFFERENTIAL EQUATION | 1 |

GAMMA FUNCTION BETA FUNCTION | 33 |

HI HYPERGEOMETRIC FUNCTION | 52 |

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### Common terms and phrases

2n+l ascending powers Bessel Functions Beta Function coefficient of tn coefficient of xp+1 Confluent Hypergeometric Function convergent Corollary defined denote differential equation dropping C0 dx dx dz2 dz Equating the coefficient Equating to zero Equation in Series erfx Example expansion Frobenius method Gamma function gives Hence the integral Hermite Polynomial hypergeometric equation hypergeometric series indicial equation integrating with respect Jn(x Laguerre polynomial large values left hand side Legendre equation Legendre polynomials Legendre's associated equation Legendre's equation Ln(x lowest degree term Multiplying both sides obtain ordinary point Pn(x Pn(x)dx polynomial of degree positive integer power series Prove recurrence relation regular singularity right hand side roots satisfies the differential satisfies the equation series of Legendre series solution series the differential Similarly singular point Solve in series Substituting these results Taylor series term containing theorem Whittaker functions write zero the coefficient