Difference between revisions of "Book:Milton Abramowitz/Handbook of mathematical functions"
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[http://people.math.sfu.ca/~cbm/aands/intro.htm#006 Hosted by Simon Fraser University]<br /> | [http://people.math.sfu.ca/~cbm/aands/intro.htm#006 Hosted by Simon Fraser University]<br /> | ||
[http://www.iopb.res.in/~somen/abramowitz_and_stegun/ Hosted by Institute of Physics, Bhubaneswar]<br /> | [http://www.iopb.res.in/~somen/abramowitz_and_stegun/ Hosted by Institute of Physics, Bhubaneswar]<br /> | ||
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[http://www.math.hkbu.edu.hk/support/aands/ Hong Kong Baptist University]<br /> | [http://www.math.hkbu.edu.hk/support/aands/ Hong Kong Baptist University]<br /> | ||
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:::[[E|$4.1.16$]] | :::[[E|$4.1.16$]] | ||
:::[[E is limit of (1+1/n)^n|$4.1.17$]] | :::[[E is limit of (1+1/n)^n|$4.1.17$]] | ||
| − | :::$4. | + | :::[[Logarithm base a|$4.1.18$]] |
| − | :::$4. | + | :::[[Log base a in terms of logarithm base b|$4.1.19$]] |
| − | :::$4. | + | :::[[Log a(z)=1/log b(a)|$4.1.20$]] |
| − | :::$4. | + | :::[[Log e(z)=log(z)|$4.1.21$]] |
| − | :::$4. | + | :::[[Log 10(z)=log(z)/log(10)|$4.1.22$]] (and [[Log 10(z)=log 10(e)log(z)|$4.1.22$]]) |
| − | :::$4. | + | :::[[Log(z)=log(10)log 10(z)|$4.1.23$]] |
:::[[Taylor series of log(1+z)|$4.1.24$]] | :::[[Taylor series of log(1+z)|$4.1.24$]] | ||
:::[[Series for log(z) for Re(z) greater than 1/2|$4.1.25$]] | :::[[Series for log(z) for Re(z) greater than 1/2|$4.1.25$]] | ||
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:::[[Laurent series for log((z+1)/(z-1)) for absolute value of z greater than 1|$4.1.28$]] | :::[[Laurent series for log((z+1)/(z-1)) for absolute value of z greater than 1|$4.1.28$]] | ||
:::[[Series for log(z+a) for positive a and Re(z) greater than -a|$4.1.29$]] | :::[[Series for log(z+a) for positive a and Re(z) greater than -a|$4.1.29$]] | ||
| + | :::[[Limit of log(x)/x^a=0|$4.1.30$]] | ||
| + | :::[[Limit of x^a log(x)=0|$4.1.31$]] | ||
| + | :::[[Euler-Mascheroni constant|$4.1.32$]] | ||
| + | :::[[X/(1+x) less than log(1+x)|$4.1.33$]] (and [[Log(1+x) less than x|$4.1.33$]]) | ||
| + | :::[[X less than -log(1-x)|$4.1.34$]] (and [[-log(1-x) less than x/(1-x)|$4.1.34$]]) | ||
| + | :::[[Abs(log(1-x)) less than 3x/2|$4.1.35$]] | ||
| + | :::[[Log(x) less than or equal to x-1|$4.1.36$]] | ||
| + | :::[[Log(x) less than or equal to n(x^(1/n)-1)|$4.1.37$]] | ||
| + | :::[[Abs(log(1+z)) less than or equal to -log(1-abs(z))|$4.1.38$]] | ||
| + | :::[[Log(1+z) as continued fraction|$4.1.39$]] | ||
| + | :::[[Log((1+z)/(1-z)) as continued fraction|$4.1.40$]] | ||
| + | :::---------- | ||
| + | :::[[Derivative of the logarithm|$4.1.46$]] | ||
| + | :::[[Nth derivative of logarithm|$4.1.47$]] | ||
| + | :::[[Logarithm|$4.1.48$]] | ||
| + | :::[[Antiderivative of the logarithm|$4.1.49$]] | ||
| + | :::[[Integral of (z^n)log(z)dz=(z^(n+1)/(n+1))log(z)-z^(n+1)/(n+1)^2 for integer n neq -1|$4.1.50$]] | ||
::4.2. Exponential Function | ::4.2. Exponential Function | ||
:::[[Exponential|$4.2.1$]] | :::[[Exponential|$4.2.1$]] | ||
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:::[[Exponential of logarithm|$4.2.4$]] | :::[[Exponential of logarithm|$4.2.4$]] | ||
:::[[Derivative of the exponential function|$4.2.5$]] | :::[[Derivative of the exponential function|$4.2.5$]] | ||
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:::[[E^(-x/(1-x)) is less than 1-x is less than e^(-x) for nonzero real x less than 1|$4.2.29$]] | :::[[E^(-x/(1-x)) is less than 1-x is less than e^(-x) for nonzero real x less than 1|$4.2.29$]] | ||
:::[[E^x is greater than 1+x for nonzero real x|$4.2.30$]] | :::[[E^x is greater than 1+x for nonzero real x|$4.2.30$]] | ||
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:::[[Period of cosh|$4.5.14$]] | :::[[Period of cosh|$4.5.14$]] | ||
:::[[Period of tanh|$4.5.15$]] | :::[[Period of tanh|$4.5.15$]] | ||
| + | :::[[Pythagorean identity for sinh and cosh|$4.5.16$]] | ||
| + | :::[[Pythagorean identity for tanh and sech|$4.5.17$]] | ||
| + | :::[[Pythagorean identity for coth and csch|$4.5.18$]] | ||
| + | :::[[Sum of cosh and sinh|$4.5.19$]] | ||
| + | :::[[Difference of cosh and sinh|$4.5.20$]] | ||
| + | :::[[Sinh is odd|$4.5.21$]] | ||
| + | :::[[Cosh is even|$4.5.22$]] | ||
| + | :::[[Tanh is odd|$4.5.23$]] | ||
| + | :::[[Sinh of a sum|$4.5.24$]] | ||
| + | :::[[Cosh of a sum|$4.5.25$]] | ||
| + | :::[[Tanh of a sum|$4.5.26$]] | ||
| + | :::[[Coth of a sum|$4.5.27$]] | ||
| + | :::[[Halving identity for sinh|$4.5.28$]] | ||
| + | :::[[Halving identity for cosh|$4.5.29$]] | ||
| + | :::[[Halving identity for tangent (1)|$4.5.30$]] (and [[Halving identity for tangent (2)|$4.5.30$]] and [[Halving identity for tangent (1)|$4.5.30$]] | ||
| + | :::[[Doubling identity for sinh (1)|$4.5.31$]] (and [[Doubling identity for sinh (2)|$4.5.31$]] | ||
| + | :::[[Doubling identity for cosh (1)|$4.5.32$]] (and [[Doubling identity for cosh (2)|$4.5.32$]] and [[Doubling identity for cosh (3)|$4.5.32$) | ||
::4.6. Inverse Hyperbolic Functions | ::4.6. Inverse Hyperbolic Functions | ||
::4.7. Use and Extension of the Tables | ::4.7. Use and Extension of the Tables | ||
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:::[[Logarithmic integral|$5.1.3$]] | :::[[Logarithmic integral|$5.1.3$]] | ||
:::[[Exponential integral E|$5.1.4$]] | :::[[Exponential integral E|$5.1.4$]] | ||
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:::[[Symmetry relation of exponential integral E|$5.1.13$]] | :::[[Symmetry relation of exponential integral E|$5.1.13$]] | ||
:::[[Recurrence relation of exponential integral E|$5.1.14$]] | :::[[Recurrence relation of exponential integral E|$5.1.14$]] | ||
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:::[[Gamma|$6.1.1$]] | :::[[Gamma|$6.1.1$]] | ||
:::[[Gauss' formula for gamma function|$6.1.2$]] | :::[[Gauss' formula for gamma function|$6.1.2$]] | ||
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::6.2. Beta Function | ::6.2. Beta Function | ||
:::[[Beta|$6.2.1$]] (and [[Beta in terms of power of t over power of (1+t)|$6.2.1$]] and [[Beta in terms of sine and cosine|$6.2.1$]]) | :::[[Beta|$6.2.1$]] (and [[Beta in terms of power of t over power of (1+t)|$6.2.1$]] and [[Beta in terms of sine and cosine|$6.2.1$]]) | ||
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:::[[Polygamma reflection formula|$6.4.7$]] | :::[[Polygamma reflection formula|$6.4.7$]] | ||
:::[[Polygamma multiplication formula|$6.4.8$]] | :::[[Polygamma multiplication formula|$6.4.8$]] | ||
| − | :::$6.4.9$ | + | :::[[Series for polygamma in terms of Riemann zeta|$6.4.9$]] |
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::6.5. Incomplete Gamma Function | ::6.5. Incomplete Gamma Function | ||
::6.6. Incomplete Beta Function | ::6.6. Incomplete Beta Function | ||
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:::[[Erfc|$7.1.2$]] | :::[[Erfc|$7.1.2$]] | ||
:::[[Two-sided inequality for e^(x^2) integral from x to infinity e^(-t^2) dt for non-negative real x|$7.1.3$]] | :::[[Two-sided inequality for e^(x^2) integral from x to infinity e^(-t^2) dt for non-negative real x|$7.1.3$]] | ||
| − | ::: | + | :::------------- |
:::[[Taylor series for error function|$7.1.5$]] | :::[[Taylor series for error function|$7.1.5$]] | ||
| − | ::: | + | :::---------- |
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:::[[Error function is odd|$7.1.9$]] | :::[[Error function is odd|$7.1.9$]] | ||
:::[[Erf of conjugate is conjugate of erf|$7.1.10$]] | :::[[Erf of conjugate is conjugate of erf|$7.1.10$]] | ||
| − | ::: | + | :::----------- |
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:::[[Continued fraction for 2e^(z^2) integral from z to infinity e^(-t^2) dt for positive Re(z)|$7.1.14$]] | :::[[Continued fraction for 2e^(z^2) integral from z to infinity e^(-t^2) dt for positive Re(z)|$7.1.14$]] | ||
:::[[Continued fraction for 1/sqrt(pi) integral from -infinity to infinity of e^(-t^2)/(z-t) dt|$7.1.15$]] | :::[[Continued fraction for 1/sqrt(pi) integral from -infinity to infinity of e^(-t^2)/(z-t) dt|$7.1.15$]] | ||
:::[[Limit of erf when z approaches infinity and the modulus of arg(z) is less than pi/4|$7.1.16$]] | :::[[Limit of erf when z approaches infinity and the modulus of arg(z) is less than pi/4|$7.1.16$]] | ||
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::7.2. Repeated Integrals of the Error Function | ::7.2. Repeated Integrals of the Error Function | ||
::7.3. Fresnel Integrals | ::7.3. Fresnel Integrals | ||
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:9. Bessel Functions of Integer Order | :9. Bessel Functions of Integer Order | ||
::9.1. Definitions and Elementary Properties | ::9.1. Definitions and Elementary Properties | ||
| − | ::: | + | :::----------- |
:::[[Bessel Y|$9.1.2$]] | :::[[Bessel Y|$9.1.2$]] | ||
:::[[Hankel H (1)|$9.1.3$]] (and [[Hankel H (1) in terms of csc and Bessel J|$9.1.3$]]) | :::[[Hankel H (1)|$9.1.3$]] (and [[Hankel H (1) in terms of csc and Bessel J|$9.1.3$]]) | ||
:::[[Hankel H (2)|$9.1.4$]] (and [[Hankel H (2) in terms of csc and Bessel J|$9.1.4$]]) | :::[[Hankel H (2)|$9.1.4$]] (and [[Hankel H (2) in terms of csc and Bessel J|$9.1.4$]]) | ||
:::[[Relationship between Bessel J sub n and Bessel J sub -n|$9.1.5$]] (and [[Relationship between Bessel Y sub n and Bessel Y sub -n|$9.1.5$]]) | :::[[Relationship between Bessel J sub n and Bessel J sub -n|$9.1.5$]] (and [[Relationship between Bessel Y sub n and Bessel Y sub -n|$9.1.5$]]) | ||
| − | ::: | + | :::----------- |
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:::[[Bessel J|$9.1.10$]] | :::[[Bessel J|$9.1.10$]] | ||
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:::[[Derivative of Bessel J with respect to its order|$9.1.64$]] | :::[[Derivative of Bessel J with respect to its order|$9.1.64$]] | ||
:::[[Derivative of Bessel Y with respect to its order|$9.1.65$]] | :::[[Derivative of Bessel Y with respect to its order|$9.1.65$]] | ||
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::9.2. Asymptotic Expansions for Large Arguments | ::9.2. Asymptotic Expansions for Large Arguments | ||
::9.3. Asymptotic Expansions for Large Orders | ::9.3. Asymptotic Expansions for Large Orders | ||
| Line 407: | Line 278: | ||
:::[[Integral of Bessel J for nu=n+1|$11.1.5$]] | :::[[Integral of Bessel J for nu=n+1|$11.1.5$]] | ||
:::[[Integral of Bessel J for nu=1|$11.1.6$]] | :::[[Integral of Bessel J for nu=1|$11.1.6$]] | ||
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::11.2 Repeated Integrals of $J_n(z)$ and $K_0(z)$ | ::11.2 Repeated Integrals of $J_n(z)$ and $K_0(z)$ | ||
::11.3 Reduction Formulas for Indefinite Integrals | ::11.3 Reduction Formulas for Indefinite Integrals | ||
| Line 438: | Line 284: | ||
:12. Struve Functions and Related Functions | :12. Struve Functions and Related Functions | ||
::12.1 Struve Function $\mathbf{H}_{\nu}(z)$ | ::12.1 Struve Function $\mathbf{H}_{\nu}(z)$ | ||
| + | :::-------------------- | ||
| + | :::[[Struve function|$12.1.3$]] | ||
| + | :::-------------------- | ||
| + | :::[[Integral representation of Struve function|$12.1.6$]] | ||
| + | :::[[Integral representation of Struve function (2)|$12.1.7$]] | ||
| + | :::[[Integral representation of Struve function (3)|$12.1.8$]] | ||
| + | :::[[Recurrence relation for Struve fuction|$12.1.9$]] | ||
| + | :::[[Recurrence relation for Struve function (2)|$12.1.10$]] | ||
| + | :::[[Derivative of Struve H0|$12.1.11$]] | ||
| + | :::[[D/dz(z^(nu)H (nu))=z^(nu)H (nu-1)|$12.1.12$]] | ||
| + | :::[[D/dz(z^(-nu)H (nu))=1/(sqrt(pi)2^(nu)Gamma(nu+3/2))-z^(-nu)H (nu+1)|$12.1.13$]] | ||
| + | :::[[H (nu)(x) geq 0 for x gt 0 and nu geq 1/2|$12.1.14$]] | ||
| + | :::[[H (-(n+1/2))(z)=(-1)^n J (n+1/2)(z) for integer n geq 0|$12.1.15$]] | ||
| + | :::[[H (1/2)(z)=sqrt(2/(pi z))(1-cos(z))|$12.1.16$]] | ||
| + | :::[[H (3/2)(z)=sqrt(z/(2pi))(1+2/z^2)-sqrt(2/(pi z))(sin(z)+cos(z)/z)|$12.1.17$]] | ||
::12.2 Modified Struve Functions $\mathbf{L}_{\nu}(z)$ | ::12.2 Modified Struve Functions $\mathbf{L}_{\nu}(z)$ | ||
::12.3 Anger and Weber Functions | ::12.3 Anger and Weber Functions | ||
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:::[[Relationship between Anger function and Weber function|$12.3.4$]]<br /> | :::[[Relationship between Anger function and Weber function|$12.3.4$]]<br /> | ||
:::[[Relationship between Weber function and Anger function|$12.3.5$]]<br /> | :::[[Relationship between Weber function and Anger function|$12.3.5$]]<br /> | ||
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::12.4 Use and Extension of the Tables | ::12.4 Use and Extension of the Tables | ||
:13. Confluent Hypergeometric Functions | :13. Confluent Hypergeometric Functions | ||
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:::[[2F1(1/2,1;3/2;-z^2)=arctan(z)/z|$15.1.5$]] | :::[[2F1(1/2,1;3/2;-z^2)=arctan(z)/z|$15.1.5$]] | ||
:::[[2F1(1/2,1/2;3/2;z^2)=arcsin(z)/z|$15.1.6$]] | :::[[2F1(1/2,1/2;3/2;z^2)=arcsin(z)/z|$15.1.6$]] | ||
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::15.2 Differentiation Formulas and Gauss' Relations for Contiguous Functions | ::15.2 Differentiation Formulas and Gauss' Relations for Contiguous Functions | ||
::15.3 Integral Representations and Transformation Formulas | ::15.3 Integral Representations and Transformation Formulas | ||
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::17.2 Canonical Forms | ::17.2 Canonical Forms | ||
::17.3 Complete Elliptic Integrals of the First and Second Kinds | ::17.3 Complete Elliptic Integrals of the First and Second Kinds | ||
| + | :::[[Elliptic K|$17.3.1$]] | ||
| + | :::---------- | ||
| + | :::[[K(m)=(pi/2)2F1(1/2,1/2;1;m)|$17.3.9$]] | ||
| + | :::[[E(m)=(pi/2)2F1(-1/2,1/2;1;m)|$17.3.10$]] | ||
::17.4 Incomplete Elliptic Integrals of the First and Second Kinds | ::17.4 Incomplete Elliptic Integrals of the First and Second Kinds | ||
::17.5 Landen's Transformation | ::17.5 Landen's Transformation | ||
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:26. Probability Functions | :26. Probability Functions | ||
:27. Miscellaneous Functions | :27. Miscellaneous Functions | ||
| + | ::------- | ||
| + | ::[[Dilogarithm|$27.7.2$]] | ||
:28. Scales of Notation | :28. Scales of Notation | ||
:29. Laplace Transforms | :29. Laplace Transforms | ||
| + | ::29.1 Definition of the Laplace Transform | ||
| + | :::[[Laplace transform|$29.1.1$]] | ||
:Subject Index | :Subject Index | ||
Latest revision as of 05:08, 21 December 2017
Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions with formulas, graphs, and mathematical tables
Published $1964$, Dover Publications
- ISBN 0-486-61272-4.
Online mirrors
Hosted by Simon Fraser University
Hosted by Institute of Physics, Bhubaneswar
Hong Kong Baptist University
BiBTeX
@book {MR0167642,
AUTHOR = {Abramowitz, Milton and Stegun, Irene A.},
TITLE = {Handbook of mathematical functions with formulas, graphs, and
mathematical tables},
SERIES = {National Bureau of Standards Applied Mathematics Series},
VOLUME = {55},
PUBLISHER = {For sale by the Superintendent of Documents, U.S. Government
Printing Office, Washington, D.C.},
YEAR = {1964},
PAGES = {xiv+1046},
}
Contents
- Preface
- Foreword
- Introduction
- 1. Mathematical Constants
- 2. Physical Constants and Conversion Factors
- 3. Elementary Analytical Methods
- 3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometric Progressions; Arithmetic, Geometric, Harmonic and Generalized Means
- 3.2. Inequalities
- 3.3. Rules for Differentiation and Integration
- 3.4. Limits, Maxima and Minima
- 3.5. Absolute and Relative Errors
- 3.6. Infinite Series
- 3.7. Complex Numbers and Functions
- 3.8. Algebraic Equations
- 3.9. Successive Approximation Methods
- 3.10. Theorems on Continued Fractions
- 3.11. Use and Extension of the Tables
- 3.12. Computing Techniques
- 4. Elementary Transcendental Functions: Logarithmic, Exponential, Circular and Hyperbolic Functions
- 4.1. Logarithmic Function
- $4.1.1$
- $4.1.2$
- $4.1.3$
- $4.1.4$
- $4.1.5$
- $4.1.6$
- $4.1.7$
- $4.1.8$
- $4.1.9$
- $4.1.10$
- $4.1.11$
- $4.1.12$
- $4.1.13$
- $4.1.14$
- $4.1.15$ (also $4.1.15$)
- $4.1.16$
- $4.1.17$
- $4.1.18$
- $4.1.19$
- $4.1.20$
- $4.1.21$
- $4.1.22$ (and $4.1.22$)
- $4.1.23$
- $4.1.24$
- $4.1.25$
- $4.1.26$
- $4.1.27$
- $4.1.28$
- $4.1.29$
- $4.1.30$
- $4.1.31$
- $4.1.32$
- $4.1.33$ (and $4.1.33$)
- $4.1.34$ (and $4.1.34$)
- $4.1.35$
- $4.1.36$
- $4.1.37$
- $4.1.38$
- $4.1.39$
- $4.1.40$
- ----------
- $4.1.46$
- $4.1.47$
- $4.1.48$
- $4.1.49$
- $4.1.50$
- 4.2. Exponential Function
- 4.3. Circular Functions
- 4.4. Inverse Circular Functions
- 4.5. Hyperbolic Functions
- $4.5.1$
- $4.5.2$
- $4.5.3$
- $4.5.4$
- $4.5.5$
- $4.5.6$
- $4.5.7$
- $4.5.8$
- $4.5.9$
- $4.5.10$
- $4.5.11$
- $4.5.12$
- $4.5.13$
- $4.5.14$
- $4.5.15$
- $4.5.16$
- $4.5.17$
- $4.5.18$
- $4.5.19$
- $4.5.20$
- $4.5.21$
- $4.5.22$
- $4.5.23$
- $4.5.24$
- $4.5.25$
- $4.5.26$
- $4.5.27$
- $4.5.28$
- $4.5.29$
- $4.5.30$ (and $4.5.30$ and $4.5.30$
- $4.5.31$ (and $4.5.31$
- $4.5.32$ (and $4.5.32$ and [[Doubling identity for cosh (3)|$4.5.32$)
- 4.6. Inverse Hyperbolic Functions
- 4.7. Use and Extension of the Tables
- 4.1. Logarithmic Function
- 5. Exponential Integral and Related Functions
- 6. Gamma Function and Related Functions
- 6.1. Gamma Function
- 6.2. Beta Function
- 6.3. Psi (Digamma Function)
- 6.4. Polygamma Functions
- 6.5. Incomplete Gamma Function
- 6.6. Incomplete Beta Function
- 6.7. Use and Extension of the Tables
- 6.8. Summation of Rational Series by Means of Polygamma Functions
- 7. Error Function and Fresnel Integrals
- 8. Legendre Functions
- 8.1. Differential Equation
- 8.2. Relations Between Legendre Functions
- 8.3. Values on the Cut
- 8.4. Explicit Expressions
- 8.5. Recurrence Relations
- 8.6. Special Values
- 8.7. Trigonometric Expressions
- 8.8. Integral Representations
- 8.9. Summation Formulas
- 8.10. Asymptotic Expansions
- 8.11. Toroidal Functions
- 8.12. Conical Functions
- 8.13. Relation to Elliptic Integrals
- 8.14. Integrals
- 8.15. Use and Extension of the Tables
- 9. Bessel Functions of Integer Order
- 9.1. Definitions and Elementary Properties
- 9.2. Asymptotic Expansions for Large Arguments
- 9.3. Asymptotic Expansions for Large Orders
- 9.4. Polynomial Approximations
- 9.5. Zeros
- 9.6. Definitions and Properties
- 9.7. Asymptotic Expansions
- 9.8. Polynomial Approximations
- 9.9. Definitions and Properties
- 9.10. Asymptotic Expansions
- 9.11. Polynomial Approximations
- 9.12. Use and Extension of the Tables
- 10. Bessel Functions of Fractional Order
- 10.1 Spherical Bessel Functions
- 10.2 Modified Spherical Bessel Functions
- 10.3 Riccati-Bessel Functions
- 10.4 Airy Functions
- 10.5 Use and Extension of the Tables
- 11. Integrals of Bessel Functions
- 12. Struve Functions and Related Functions
- 12.1 Struve Function $\mathbf{H}_{\nu}(z)$
- 12.2 Modified Struve Functions $\mathbf{L}_{\nu}(z)$
- 12.3 Anger and Weber Functions
- 12.4 Use and Extension of the Tables
- 13. Confluent Hypergeometric Functions
- 13.1 Definitions of Kummer and Whittaker Functions
- 13.2 Integral Representations
- 13.3 Connections With Bessel Functions
- 13.4 Recurrence Relations and Differential Properties
- 13.5 Asymptotic Expansions and Limiting Forms
- 13.6 Special Cases
- 13.7 Zeros and Turning Values
- 13.8 Use and Extension of the Tables
- 13.9 Calculation of the Zeros and Turning Points
- 13.10 Graphing $M(a,b,x)$
- 14. Coulomb Wave Functions
- 14.1 Differential Equation, Series Expansions
- 14.2 Recurrence and Wronskian Relations
- 14.3 Integral Representations
- 14.4 Bessel Function Expansions
- 14.5 Asymptotic Expansions
- 14.6 Special Values and Asymptotic Behavior
- 14.7 Use and Extension of the Tables
- 15. Hypergeometric Functions
- 15.1 Gauss Series, Special Elementary Cases, Special Values of the Argument
- 15.2 Differentiation Formulas and Gauss' Relations for Contiguous Functions
- 15.3 Integral Representations and Transformation Formulas
- 15.4 Special Cases of $F(a,b;c;z)$, Polynomials and Legendre Functions
- 15.5 The Hypergeometric Differential Equation
- 15.6 Riemann's Differential Equation
- 15.7 Asymptotic Expansions
- 16. Jacobian Elliptic Functions and Theta Functions
- 16.1 Introduction
- 16.2 Classification of the Twelve Jacobian Elliptic Functions
- 16.3 Relation of the Jacobian Functions to the Copolor Trio
- 16.4 Calculation of the Jacobian Functions by Use of the Arithmetic-Geometric Mean (A.G.M.)
- 16.5 Special Arguments
- 16.6 Jacobian Functions when $m=0$ or $1$
- 16.7 Principal Terms
- 16.8 Change of Argument
- 16.9 Relations Between the Squares of the Functions
- 16.10 Change of Parameter
- 16.11 Reciprocal Parameter
- 16.12 Descending Landen Transformation (Gauss' Transformation)
- 16.13 Approximation in Terms of Circular Functions
- 16.14 Ascending Landen Transformation
- 16.15 Approximation in Terms of Hyperbolic Functions
- 16.16 Derivatives
- 16.17 Addition Theorems
- 16.18 Double Arguments
- 16.19 Half Arguments
- 16.20 Jacobi's Imaginary Transformation
- 16.21 Complex Arguments
- 16.22 Leading Terms of the Series in Ascending Powers of $u$
- 16.23 Series Expansion in Terms of the Nome $q$
- 16.24 Integrals of the Twelve Jacobian Elliptic Functions
- 16.25 Notation for the Integrals of the Squares of the Twelve Jacobian Elliptic Functions
- 16.26 Integrals in Terms of the Elliptic Integral of the Second Kind
- 16.27 Theta Functions; Expansions in Terms of the Nome $q$
- 16.28 Relations Between the Squares of the Theta Functions
- 16.29 Logarithmic Derivatives of the Theta Functions
- 16.30 Logarithms of Theta Functions of Sum and Difference
- 16.31 Jacobi's Notation for Theta Functions
- 16.32 Calculation of Jacobi's Theta Function $\Theta(u|m)$ by Use of the Arithmetic-Geometric Mean
- 16.33 Addition of Quarter-Periods to Jacobi's Eta and Theta Functions
- 16.34 Relation of Jacobi's Zeta Function to the Theta Functions
- 16.35 Calculation of Jacobi's Zeta Function $Z(u|m)$ by Use of the Arithmetic-Geometric Mean
- 16.36 Neville's Notation for Theta Functions
- 16.37 Expression as Infinite Products
- 16.38 Expression as Infinite Series
- 16.39 Use and Extension of the Tables
- 17. Elliptic Integrals
- 17.1 Definition of Elliptic Integrals
- 17.2 Canonical Forms
- 17.3 Complete Elliptic Integrals of the First and Second Kinds
- 17.4 Incomplete Elliptic Integrals of the First and Second Kinds
- 17.5 Landen's Transformation
- 17.6 The Process of the Arithmetic-Geometric Mean
- 17.7 Elliptic Integrals of the Third Kind
- 17.8 Use and Extension of the Tables
- 18. Weierstrass Elliptic and Related Functions
- 18.1 Definitions, Symbolism, Restrictions and Conventions
- 18.2 Homogeneity Relations, Reduction Formulas and Processes
- 18.3 Special Values and Relations
- 18.4 Addition and Multiplication Formulas
- 18.5 Series Expansions
- 18.6 Derivatives and Differential Equations
- 18.7 Integrals
- 18.8 Conformal Mapping
- 18.9 Relations with Complete Elliptic Integrals $K$ and $K'$ and Their Parameter $m$ and with Jacobi's Elliptic Functions
- 18.10 Relations with Theta Functions
- 18.11 Expressing and Elliptic Function in Terms of $\wp$ and $\wp'$
- 18.12 Case $\Delta=0$
- 18.13 Equianharmonic Case ($g_2=0,g_3=1$)
- 18.14 Lemniscatic Case ($g_2=1, g_3=0$)
- 18.15 Pseudo-Lemniscatic Case ($g_2=-1, g_3=0$)
- 18.16 Use and Extension of the Tables
- 19. Parabolic Cylinder Functions
- 20. Mathieu Functions
- 21. Spheroidal Wave Functions
- 22. Orthogonal Polynomials
- 23. Bernoulli and Euler Polynomials, Riemann Zeta Function
- 24. Combinatorial Analysis
- 24.1. Basic numbers
- 24.1.1. Binomial Coefficients
- 24.1.2. Multinomial Coefficients
- 24.1.3. Stirling Numbers of the First Kind
- 24.1.4. Stirling Numbers of the Second Kind
- 24.2. Partitions
- 24.2.1. Unrestricted Partitions
- 24.2.2. Partitions Into Distinct Parts
- 24.3. Number Theoretic Functions
- 24.3.1. The Mobius Function
- 24.3.2. The Euler Function
- 24.3.3. Divisor Functions
- 24.3.4. Primitive Roots
- References
- 24.1. Basic numbers
- 25. Numerical Interpolation, Differentiation and Integration
- 26. Probability Functions
- 27. Miscellaneous Functions
- -------
- $27.7.2$
- 28. Scales of Notation
- 29. Laplace Transforms
- 29.1 Definition of the Laplace Transform
- Subject Index
- Index of Notations
