An atlas of functions with Equator, the atlas function calculator ; over 300 diagrams in color
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| Vorheriger Titel: | Spanier, Jerome An atlas of functions |
|---|---|
| Hauptverfasser: | , , |
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
2009
|
| Ausgabe: | 2. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
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| 100 | 1 | |a Oldham, Keith B. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a An atlas of functions |b with Equator, the atlas function calculator ; over 300 diagrams in color |c Keith Oldham, Jan Myland, & Jerome Spanier |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 2009 | |
| 300 | |a XI, 748 S. |b Ill., graph. Darst. |e 1 CD-ROM (12 cm) | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 0 | 7 | |a Funktion |g Mathematik |0 (DE-588)4071510-3 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)4155008-0 |a Formelsammlung |2 gnd-content | |
| 689 | 0 | 0 | |a Funktion |g Mathematik |0 (DE-588)4071510-3 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Myland, Jan C. |e Verfasser |4 aut | |
| 700 | 1 | |a Spanier, Jerome |e Verfasser |4 aut | |
| 780 | 0 | 0 | |i 1. Auflage |a Spanier, Jerome |t An atlas of functions |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016798297&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-016798297 | |
Datensatz im Suchindex
| _version_ | 1819747083001790464 |
|---|---|
| adam_text | CONTENTS
Every chapter has sections devoted to: notation, behavior, definitions, special cases, intrarelationships, expansions,
particular values, numerical values, limits and approximations, operations of the calculus, complex argument,
generalizations, and cognate functions. In addition, each chapter has the special features itemized below its title.
PREFACE í
GENERAL CONSIDERATIONS 1
VJ What functions are. Organization of the Atlas. Notational conventions. Rules of the calculus.
X Mathematical constants. Complex numbers. Pulse functions. Series of powers of natural numbers
2
THE CONSTANT FUNCTION c 13
Mathematical constants. Complex numbers. Pulse functions. Series of powers of natural numbers.
THE FACTORIAL FUNCTION ç ! 21
Double and triple factorial functions. Combinatorics. Stirling numbers of the second kind.
THE ZETA NUMBERS AND RELATED FUNCTIONS 29
~J Special values. Apery s constant. The Debye functions of classical physics.
4
THE BERNOULLI NUMBERS B„ 39
Dual definitions. Relationship to zeta numbers. The Euler-Maclaurin sum formulas.
THE EULER NUMBERS E,, 45
Relationship to beta numbers and Bernoulli numbers.
r THE BINOMIAL COEFFICIENTS («) 49
J Binomial expansion. Pascal s triangle. The Laplace-de Moivre formula. Multinomial coefficients.
THE LINEAR FUNCTION bx + c AND ITS RECIPROCAL 57
How to fit data to a best straight line . Errors attaching to the fitted parameters.
viii CONTENTS
O
MODIFYING FUNCTIONS 67
Selecting features of numbers. Rounding. Base conversion.
THE HEAVISIDE u(x-a) AND DIRAC ä(÷-á) FUNCTIONS 75
Window and other discontinuous functions. The comb function. Green s functions.
-é /ë THE INTEGER POWERS x» AND (bx + c)n 81
X J Summing power series. Euler transformation. Transformations through lozenge diagrams.
11
12
THE SQUARE-ROOT FUNCTION yfbx + c AND ITS RECIPROCAL 95
Behavior of the semiparabolic function in the complex plane. The parabola and its geometry.
THE NONINTEGER POWER xv 103
Behavior in four quadrants. Mellin transforms. De Moivre s theorem. The fractional calculus.
THE SEMIELLIPTIC FUNCTION (b I a)-Ja2 - x2 AND ITS RECIPROCAL 113
Ellipticity. Geometric properties of the ellipse, ellipsoid, and semicircle. Superellipses.
é ë THE (b I a)y/x2 ± a2 FUNCTIONS AND THEIR RECIPROCALS 121
X T Vertical, horizontal and diagonal varieties of hyperbolas. Operations that interrelate functions graphically.
é ^ THE QUADRATIC FUNCTION ax2 + bx + c AND ITS RECIPROCAL 131
X «_/ Zeros, real and complex. The root-quadratic function. Conic sections. Trajectory of a projectile.
-i r THE CUBIC FUNCTION x3 + ax2 + bx + c 139
X J Zeros of cubics and quartics. Joining the dots with sliding cubics and cubic splines.
17
18
POLYNOMIAL FUNCTIONS 147
Finding zeros. Rational functions. Partial fractions. Polynomial optimization and regression.
THE POCHHAMMER POLYNOMIALS (x)„ 159
Stirling numbers of the first kind. Hypergeometric functions.
é q THE BERNOULLI POLYNOMIALS Bn(x) 175
X Zs Sums of monotonie power series.
THE EULER POLYNOMIALS En(x) 181
Sums of alternating power series.
THE LEGENDRE POLYNOMIALS ?n(x) 187
Orthogonality. The Legendre differential equation and its other solution, the Q„(x) function.
rsr* THE CHEBYSHEV POLYNOMIALS T,,(x) AND U„(x) 197
?* Z* Gegenbauer and Jacobi polynomials. Fitting data sets with discrete Chebyshev polynomials.
23
THE LAGUERRE POLYNOMIALS L„(x) 209
Associated Laguerre polynomials. Fibonacci numbers and the golden section.
25
CONTENTS ¡x
THE HERMITE POLYNOMIALS H„(x) 217
Gauss integration. The systematic solution of second-order differential equations.
THE LOGARITHMIC FUNCTION ln(x) 229
Logarithms to various bases. The dilogarithm and polylogarithms. Logarithmic integral.
rs r THE EXPONENTIAL FUNCTION exp(±x) 241
Á* J Exponential growth/decay. Self-exponential function. Exponential polynomial. Laplace transforms.
?, ? EXPONENTIAL OF POWERS exp(±xv) 255
¿* I Exponential theta functions. Various distributions; their probability and cumulative versions.
THE HYPERBOLIC COSINE cosh(x) AND SINE sinh(x) FUNCTIONS 269
Algebraic and geometric interpretations. The catenary.
THE HYPERBOLIC SECANT AND COSECANT FUNCTIONS 281
Interrelations between hyperbolic functions via similar triangles. Interesting inverse Laplace transformations.
THE HYPERBOLIC TANGENT AND COTANGENT FUNCTIONS 289
The Langevin function, important in the theory of electrical or magnetic dipoles.
THE INVERSE HYPERBOLIC FUNCTIONS 297
Synthesis from reciprocal linear functions. Logarithmic equivalence. Expansion as hypergeometric functions.
THE COSINE cos(x) AND SINE sin(x) FUNCTIONS 309
Sinusoids. Periodicity, frequency, phase and amplitude. Fourier transforms. Clausen s integral.
THE SECANT sec(x) AND COSECANT csc(x) FUNCTIONS 329
Interrelationships between circular functions via similar triangles. The Gudermannian function and its inverse.
THE TANGENT tan(x) AND COTANGENT cot(x) FUNCTIONS 339
Tangent and cotangent roots. Utility of half-argument formulas. Rules of trigonometry.
THE INVERSE CIRCULAR FUNCTIONS 351
Synthesis from reciprocal linear functions. Two-dimensional coordinate systems and scale factors.
PERIODIC FUNCTIONS 367
Expansions in sines and cosines. Euler s formula and Parseval s relationship. Waveforms.
29
30
31
34
35
36
^ ? THE EXPONENTIAL INTEGRALS Ei(x) AND Ein(x) 375
¦D I Cauchy limits. Functions defined as indefinite integrals. Table of popular integrals.
38
SINE AND COSINE INTEGRALS 385
Entire cosine versions. Auxiliary sine and cosine integrals.
- Q THE FRESNEL INTEGRALS C(x) AND S(x) 395
Jj 27 Böhmer integrals. Auxiliary7 Fresnel integrals. Curvatures and lengths of plane curves. Cornu s spiral.
CONTENTS
ë r THE ERROR FUNCTION erf(x) AND ITS COMPLEMENT erfc(x) 405
é J Inverse error function. Repeated integrals. Normal probability. Random numbers. Monte Carlo.
ë ë THE exp(jc)erfc(4x) AND RELATED FUNCTIONS 417
é A Properties in the complex plane and the Voigt function.
Properties in the complex plan
42
DAWSON S INTEGRAL daw(x) 427
The closely related erfi function. Intermediacy to exponentials. Gaussian integrals of complex argument.
THE GAMMA FUNCTION I» 435
Gauss-Legendre formula. Complete beta function. Function synthesis and basis hypergeometric functions.
Á ë THE DIGAMMA FUNCTION ø(í) 449
Ô ô Polygamma functions. Bateman s G function and its derivatives. Sums of reciprocal linear functions.
45
THE INCOMPLETE GAMMA FUNCTIONS 461
The Mittag-Leffler, or generalized exponential, function.
ë r THE PARABOLIC CYLINDER FUNCTION Dv(x) 471
é J Three-dimensional coordinate systems. The Laplacian, separability, and an exemplary application.
47
THE KUMMER FUNCTION M(a,c,x) 485
The confluent hypergeometric differential equation. Kummer s transformation. Zeros.
THE TRICOMI FUNCTION O(a,c,x) 497
Numbers of zeros and extrema. The two Whittaker functions. Bateman s confluent function.
THE MODIFIED BESSEL FUNCTIONS L,(x) OF INTEGER ORDER 507
Bessel s modified differential equation. Cylinder functions generally and their classification.
?- r THE MODIFIED BESSEL FUNCTION Iv(x) OF ARBITRARY ORDER 519
Simplifications when the order is a multiple of )/,, X, or ?^ . Auxiliary cylinder functions.
1
THE MACDONALD FUNCTION Kv(x) 527
Alternative solutions to Bessel s modified differential equation. Spherical Macdonald functions.
rr* THE BESSEL FUNCTIONS Jn(x) OF INTEGER ORDER 537
«_/ A* Zeros, extrema, and their associated values. Miller s method. The Newton-Raphson root-finding method.
THE BESSEL FUNCTION J,(x) OF ARBITRARY ORDER 553
The Bessel-Clifford equation. Hankel transforms. Neumann series. Discontinuous Bessel integrals.
THE NEUMANN FUNCTION Y,(x) 567
Behavior close to zero argument. Hankel functions. Asymptotic expansions of cylinder functions.
THE KELVIN FUNCTIONS 577
Complex-plane relationships to Bessel functions.
54
55
CONTENTS
r s THE AIRY FUNCTIONS Ai(x) AND Bi(x) 585
«_/ J Hyperbolic/circular chimera. Airy s differential equation. Auxiliary Airy functions. Airy derivatives.
57
THE STRUVE FUNCTION hv(x) 593
Kinship with Neumann functions. The modified Struve function.
THE INCOMPLETE BETA FUNCTION Â(í,ì,÷) 603
Role as a hypergeometric function. Integrals of circular and hyperbolic functions raised to an arbitrary power.
?- Q THE LEGENDRE FUNCTIONS P,(x) AND Qv(jc) 611
*J JS The associated Legendre functions. Solving the Laplace equation in spherical coordinates.
60
THE GAUSS HYPERGEOMETRIC FUNCTION ¥(a,b,c,x) 627
Plethora of special cases. Contiguity relationships. Linear transformations.
r Ë THE COMPLETE ELLIPTIC INTEGRALS K(k) AND E(k) 637
J X The third kind of complete elliptic integral. Means. The elliptic nome. Theta functions of various kinds.
THE INCOMPLETE ELLIPTIC INTEGRALS F(fr,q ) AND E(fc, p) 653
The Landen transformations. Ð(í?Á-,ö). Integrals of reciprocal cubic functions. Romberg integration.
THE JACOBIAN ELLIPTIC FUNCTIONS 671
Trigonometric interpretation. Circular/hyperbolic intermediacy. Double periodicity in the complex plane.
63
s~a THE HURWITZ FUNCTION æ(í,éÞ 685
U é Bivariate eta function. The Lerch function. Weyl differintegration and its application to periodic functions.
APPENDIX A: USEFUL DATA 697
SI units and prefixes. Universal constants. Terrestrial constants and standards. The Greek alphabet.
APPENDIX B: BIBLIOGRAPHY 703
Cited sources and supporting publications. Books and web sources but not original research articles.
APPENDIX C: EQUATOR, THE ATLAS FUNCTION CALCULATOR 705
Disk installation. Basic operations and additional features. Input/output formats. Accuracy. Keywords.
SYMBOL INDEX 723
Notation used here and elsewhere.
SUBJECT INDEX 735
A comprehensive directory to the topics in this Atlas.
|
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| bvnumber | BV035130796 |
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| ctrlnum | (OCoLC)633467755 (DE-599)BVBBV035130796 |
| discipline | Mathematik |
| edition | 2. ed. |
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| genre | (DE-588)4155008-0 Formelsammlung gnd-content |
| genre_facet | Formelsammlung |
| id | DE-604.BV035130796 |
| illustrated | Illustrated |
| indexdate | 2024-12-23T21:15:41Z |
| institution | BVB |
| isbn | 9780387488066 9780387488073 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016798297 |
| oclc_num | 633467755 |
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| owner_facet | DE-20 DE-29T DE-523 DE-634 DE-824 DE-11 DE-521 DE-188 |
| physical | XI, 748 S. Ill., graph. Darst. 1 CD-ROM (12 cm) |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| spellingShingle | Oldham, Keith B. Myland, Jan C. Spanier, Jerome An atlas of functions with Equator, the atlas function calculator ; over 300 diagrams in color Funktion Mathematik (DE-588)4071510-3 gnd |
| subject_GND | (DE-588)4071510-3 (DE-588)4155008-0 |
| title | An atlas of functions with Equator, the atlas function calculator ; over 300 diagrams in color |
| title_auth | An atlas of functions with Equator, the atlas function calculator ; over 300 diagrams in color |
| title_exact_search | An atlas of functions with Equator, the atlas function calculator ; over 300 diagrams in color |
| title_full | An atlas of functions with Equator, the atlas function calculator ; over 300 diagrams in color Keith Oldham, Jan Myland, & Jerome Spanier |
| title_fullStr | An atlas of functions with Equator, the atlas function calculator ; over 300 diagrams in color Keith Oldham, Jan Myland, & Jerome Spanier |
| title_full_unstemmed | An atlas of functions with Equator, the atlas function calculator ; over 300 diagrams in color Keith Oldham, Jan Myland, & Jerome Spanier |
| title_old | Spanier, Jerome An atlas of functions |
| title_short | An atlas of functions |
| title_sort | an atlas of functions with equator the atlas function calculator over 300 diagrams in color |
| title_sub | with Equator, the atlas function calculator ; over 300 diagrams in color |
| topic | Funktion Mathematik (DE-588)4071510-3 gnd |
| topic_facet | Funktion Mathematik Formelsammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016798297&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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