## An Atlas of Functions: with Equator, the Atlas Function CalculatorThis book comprehensively covers several hundred functions or function families. In chapters that progress by degree of complexity, it starts with simple, integer-valued functions then moves on to polynomials, Bessel, hypergeometric and hundreds more. |

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### Contents

THE HEAVISIDE uxa AND DIRAC FUNCTIONS | 75 |

THE INTEGER POWERS xn AND bx + cn | 81 |

Summing power series Euler transformation Transformations through lozenge diagrams | 95 |

xa | 103 |

Behavior in four quadrants Mellin transforms De Moivres theorem The fractional calculus | 113 |

Ellipticity Geometric properties of the ellipse ellipsoid and semicircle Superellipses | 121 |

THE QUADRATIC FUNCTION ax2 + bx + c AND ITS RECIPROCAL | 131 |

Zeros real and complex The rootquadratic function Conic sections Trajectory of a projectile | 139 |

THE INVERSE CIRCULAR FUNCTIONS | 351 |

PERIODIC FUNCTIONS | 367 |

SINE AND COSINE INTEGRALS 385 | 384 |

THE ERROR FUNCTION erfx AND ITS COMPLEMENT erfcx | 405 |

Properties in the complex plane and the Voigt function | 427 |

THE DIGAMMA FUNCTION v | 449 |

THE PARABOLIC CYLINDER FUNCTION Dvx | 471 |

Threedimensional coordinate systems The Laplacian separability and an exemplary application | 485 |

Zeros of cubics and quartics Joining the dots with sliding cubics and cubic splines | 147 |

Finding zeros Rational functions Partial fractions Polynomial optimization and regression | 159 |

Stirling numbers of the first kind Hypergeometric functions | 175 |

Sums of monotonic power series | 181 |

THE LEGENDRE POLYNOMIALSP nx | 187 |

Orthogonality The Legendre differential equation and its other solution the Qnx function | 197 |

Gegenbauer and Jacobi polynomials Fitting data sets with discrete Chebyshev polynomials | 209 |

THE HERMITE POLYNOMIALSHnx | 217 |

THE EXPONENTIAL FUNCTION expx | 241 |

Exponential growthdecay Selfexponential function Exponential polynomial Laplace transforms | 256 |

THE HYPERBOLIC SECANT AND COSECANT FUNCTIONS 281 | 280 |

THE INVERSE HYPERBOLIC FUNCTIONS | 297 |

THE SECANT secx AND COSECANT cscx FUNCTIONS | 329 |

THE MODIFIED BESSEL FUNCTIONS Inx OF INTEGER ORDER | 507 |

THE MACDONALD FUNCTION Kvx | 527 |

THE BESSEL FUNCTION Jvx OF ARBITRARY ORDER | 553 |

Behavior close to zero argument Hankel functions Asymptotic expansions of cylinder functions | 577 |

THE AIRY FUNCTIONS Aix AND Bix | 585 |

Kinship with Neumann functions The modified Struve function | 603 |

The associated Legendre functions Solving the Laplace equation in spherical coordinates | 627 |

THE INCOMPLETEELLIPTIC INTEGRALS FkQ AND EkQ | 653 |

THE JACOBIAN ELLIPTIC FUNCTIONS 671 | 672 |

THE HURWITZ FUNCTION vu | 685 |

703 | |

SYMBOL INDEX | 723 |

### Other editions - View all

An Atlas of Functions: with Equator, the Atlas Function Calculator Keith B. Oldham,Jan Myland,Jerome Spanier No preview available - 2008 |

### Common terms and phrases

arccot arcsin arctan asymptotic Atlas of Functions behavior Bernoulli numbers binomial coefficient calculated Chebyshev polynomials circular functions COGNATE FUNCTIONS COMPLEX ARGUMENT constant coordinate cosh cubic function defined definition differential equation differintegration elliptic functions elliptic integral Equator’s Euler example expansion exponential function expressed Figure function Chapter function Section functions of Chapter Gauss hypergeometric function Gudermannian function hyperbolic functions hypergeometric function incomplete elliptic integrals incomplete gamma function indefinite integrals infinite integer integer order INTRARELATIONSHIPS inverse hyperbolic K.B. Oldham Kummer function Laguerre polynomial Laplace transforms Legendre functions LIMITS AND APPROXIMATIONS logarithm modified Bessel function n n n negative notation NUMERICAL VALUES parabolic cylinder function parameters PARTICULAR VALUES Pochhammer polynomial positive integer power series real and imaginary reciprocal reflection formula sinh Springer Science+Business Media Struve function symbol Tricomi function unity v v v variable x x x x x zero