1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
|
References
===================
The following is a non-comprehensive list of works used in the development of mpmath
or cited for examples or mathematical definitions used in this documentation.
References not listed here can be found in the source code.
.. [AbramowitzStegun] M Abramowitz & I Stegun. *Handbook of Mathematical
Functions, 9th Ed.*, Tenth Printing, December 1972,
with corrections (electronic copy:
http://people.math.sfu.ca/~cbm/aands/)
.. [Abate] Abate, J., P. Valko (2004). Multi-precision Laplace transform
inversion. *International Journal for Numerical Methods
in Engineering* 60:979-993, http://dx.doi.org/10.1002/nme.995
.. [Ainsworth] O. R. Ainsworth & L. W. Howell, "An integral representation
of the generalized Euler-Mascheroni constants", NASA
Technical Paper 2456 (1985),
http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19850014994_1985014994.pdf
.. [Bailey] D H Bailey. "Tanh-Sinh High-Precision Quadrature",
http://crd.lbl.gov/~dhbailey/dhbpapers/dhb-tanh-sinh.pdf
.. [Bellman] Bellman, R., R.E. Kalaba, J.A. Lockett (1966). *Numerical
inversion of the Laplace transform: Applications to Biology,
Economics, Engineering, and Physics*. Elsevier.
.. [BenderOrszag] C M Bender & S A Orszag. *Advanced Mathematical Methods for
Scientists and Engineers*, Springer 1999
.. [Bernoulli] The Bernoulli Number Page: http://www.bernoulli.org/
.. [BorweinBailey] J Borwein, D H Bailey & R Girgensohn. *Experimentation in
Mathematics - Computational Paths to Discovery*,
A K Peters, 2003
.. [BorweinBorwein] J Borwein & P B Borwein. *Pi and the AGM: A Study in
Analytic Number Theory and Computational Complexity*,
Wiley 1987
.. [BorweinTanhSinh] Borwein, Jonathan Michael and Lingyun Ye. “Quadratic
Convergence of the Tanh-sinh Quadrature Rule.” (2006).
https://web.archive.org/web/20080221230631/http://users.cs.dal.ca/~jborwein/tanh-sinh.pdf
.. [BorweinZeta] P Borwein. "An Efficient Algorithm for the Riemann Zeta
Function", http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
.. [Brent79] R. P. Brent, On the Zeros of the Riemann Zeta Function in the
Critical Strip, Math. Comp. 33 (1979) 1361--1372
.. [Brent86] R. P. Brent, J. van de Lune, H. J. J. te Riele, D. T. Winter,
'On the Zeros of the Riemann Zeta Function in the Critical
Strip. II', Math. Comp. 39 (1982) 681--688.
.. [Buhring] Wolfgang Buhring, "Generalized Hypergeometric Functions at Unit
Argument", Proc. Amer. Math. Soc., Vol. 114, No. 1 (Jan. 1992),
pp.145-153
.. [CabralRosetti] L G Cabral-Rosetti & M A Sanchis-Lozano. "Appell Functions
and the Scalar One-Loop Three-point Integrals in Feynman
Diagrams". http://arxiv.org/abs/hep-ph/0206081
.. [Carlson] B C Carlson. "Numerical computation of real or complex elliptic
integrals". http://arxiv.org/abs/math/9409227v1
.. [Coffey] M. W. Coffey, "The Stieltjes constants, their relation to the
`\eta_j` coefficients, and representation of the Hurwitz zeta
function", arXiv:0706.0343v1 http://arxiv.org/abs/0706.0343
.. [Cohen] Cohen, A.M. (2007). Numerical Methods for Laplace Transform
Inversion, Springer.
.. [Corless] R M Corless et al. "On the Lambert W function", Adv. Comp.
Math. 5 (1996) 329-359.
http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf
.. [Crandall] Richard Crandall, "Note on fast polylogarithm computation"
http://www.reed.edu/physics/faculty/crandall/papers/Polylog.pdf
.. [Davies] Davies, B. (2005). *Integral Transforms and their Applications*,
Third Edition. Springer.
.. [Davies79] Davies, B., B. Martin (1979). Numerical inversion of the Laplace
transform: a survey and comparison of methods. *Journal of
Computational Physics* 33:1-32,
http://dx.doi.org/10.1016/0021-9991(79)90025-1
.. [Duffy93] Duffy, D.G. (1993). On the numerical inversion of Laplace
transforms: Comparison of three new methods on characteristic
problems from applications. *ACM Transactions on Mathematical
Software* 19(3):333-359, http://dx.doi.org/10.1145/155743.155788
.. [Duffy98] Duffy, D.G. (1998). Advanced Engineering Mathematics, CRC Press.
.. [DLMF] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/
.. [Froberg] Carl-Erik Froberg, "On the prime zeta function", BIT 8 (1968),
pp. 187-202.
.. [Glasserman] P. Glasserman, J. Ruiz-Mata (2006). Computing the credit loss
distribution in the Gaussian copula model: a comparison of
methods. *Journal of Credit Risk* 2(4):33-66,
10.21314/JCR.2006.057
.. [Golub] golub, "some modified matrix eigenvalue problems", siam review
15, p. 318-334 (1973)
.. [GolubWelsch] golub and welsch, "calculations of gaussian quadrature
rules", mathematics of computation 23, p. 221-230 (1969)
.. [Gourdon] Xavier Gourdon & Pascal Sebah, The Euler constant: gamma
http://numbers.computation.free.fr/Constants/Gamma/gamma.pdf
.. [GradshteynRyzhik] I S Gradshteyn & I M Ryzhik, A Jeffrey & D Zwillinger
(eds.), *Table of Integrals, Series and Products*,
Seventh edition (2007), Elsevier
.. [GravesMorris] P R Graves-Morris, D E Roberts & A Salam. "The epsilon
algorithm and related topics", *Journal of Computational
and Applied Mathematics*, Volume 122, Issue 1-2
(October 2000)
.. [Homeier] H.H.H. Homeier - "Scalar Levin-Type Sequence Transformations"
arXiv:math/0005209
.. [Hoog] de Hoog, F., J. Knight, A. Stokes (1982). An improved method for
numerical inversion of Laplace transforms. *SIAM Journal of
Scientific and Statistical Computing* 3:357-366,
http://dx.doi.org/10.1137/0903022
.. [Kresser] Numerical Methods for General and Structured Eigenvalue Problems
.. [Kuhlman] Kuhlman, K.L., (2013). Review of Inverse Laplace Transform
Algorithms for Laplace-Space Numerical Approaches,
*Numerical Algorithms*, 63(2):339-355.
http://dx.doi.org/10.1007/s11075-012-9625-3
.. [Lune84] J. van de Lune, 'Sums of Equal Powers of Positive Integers',
Dissertation, Vrije Universiteit te Amsterdam, Centrum voor
Wiskunde en Informatica, Amsterdam, 1984.
.. [Lune86] J. van de Lune, H. J. J. te Riele, 'On the Zeros of the Riemann
Zeta Function in the Critical Strip. III', Math. Comp. 41
(1983) 759--767.
.. [MPFR] The MPFR team. "The MPFR Library: Algorithms and Proofs",
http://www.mpfr.org/algorithms.pdf
.. [Michel] N. Michel, "Precise Coulomb wave functions for a wide
range of complex `l`, `\eta` and `z`",
http://arxiv.org/abs/physics/0702051v1
.. [OEIS] The On-Line Encyclopedia of Integer Sequences (OEIS).
.. [Sidi] A. Sidi - "Pratical Extrapolation
Methods".
.. [Slater] L J Slater. *Generalized Hypergeometric Functions*.
Cambridge University Press, 1966
.. [Spouge] J L Spouge. "Computation of the gamma, digamma, and trigamma
functions", SIAM J. Numer. Anal. Vol. 31, No. 3, pp. 931-944,
June 1994.
.. [SrivastavaKarlsson] H M Srivastava & P W Karlsson. *Multiple Gaussian
Hypergeometric Series*. Ellis Horwood, 1985.
.. [Stehfest] Stehfest, H. (1970). Algorithm 368: numerical inversion of
Laplace transforms. *Communications of the ACM* 13(1):47-49,
http://dx.doi.org/10.1145/361953.361969
.. [Stoer] Stoer, Bulirsch - Introduction to Numerical Analysis.
.. [Stroud] stroud and secrest, "gaussian quadrature formulas",
prentice-hall (1966)
.. [Talbot] Talbot, A. (1979). The accurate numerical inversion of Laplace
transforms. *IMA Journal of Applied Mathematics* 23(1):97,
http://dx.doi.org/10.1093/imamat/23.1.97
.. [Thompson] I.J. Thompson & A.R. Barnett, "Coulomb and Bessel Functions
of Complex Arguments and Order", J. Comp. Phys., vol 64, no.
2, June 1986.
.. [Trudgian] T. Trudgian, Improvements to Turing Method,
Math. Comp.
.. [Vidunas] R Vidunas. "Identities between Appell's and hypergeometric
functions". http://arxiv.org/abs/0804.0655
.. [Voros2003] A. Voros, Zeta functions for the Riemann zeros, Ann.
Institute Fourier, 53, (2003) 665--699.
.. [Voros2009] A. Voros, Zeta functions over Zeros of Zeta Functions,
Lecture Notes of the Unione Matematica Italiana, Springer, 2009.
.. [Weisstein] E W Weisstein. *MathWorld*. http://mathworld.wolfram.com/
.. [Weniger] E.J. Weniger - "Nonlinear Sequence Transformations for the
Acceleration of Convergence and the Summation of Divergent
Series" arXiv:math/0306302
.. [WhittakerWatson] E T Whittaker & G N Watson. *A Course of Modern Analysis*.
4th Ed. 1946 Cambridge University Press
.. [Widder] Widder, D. (1941). *The Laplace Transform*. Princeton.
.. [Wikipedia] *Wikipedia, the free encyclopedia*.
http://en.wikipedia.org/wiki/Main_Page
.. [WolframFunctions] Wolfram Research, Inc. *The Wolfram Functions Site*.
http://functions.wolfram.com/
|
