View source for Book:Milton Abramowitz/Handbook of mathematical functions

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{{Book|Handbook of mathematical functions with formulas, graphs, and mathematical tables|1964|Dover Publications|0-486-61272-4|Milton Abramowitz|author2 = Irene A. Stegun}}

===Online mirrors===
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[http://mintaka.sdsu.edu/faculty/wfw/ABRAMOWITZ-STEGUN/ Hosted by Bill Welsh (San Diego State University)]<br />
[http://www.math.hkbu.edu.hk/support/aands/ Hong Kong Baptist University]<br />

===BiBTeX===
<pre>@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},
}</pre>

=== 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
:::[[Constant multiple rule for derivatives|$3.3.1$]]
:::[[Sum rule for derivatives|$3.3.2$]]
:::[[Product rule for derivatives|$3.3.3$]]
:::[[Quotient rule for derivatives|$3.3.4$]]
:::[[Chain rule for derivatives|$3.3.5$]]
::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
:::[[Logarithm|$4.1.1$]]
:::[[Real and imaginary parts of log|$4.1.2$]]
:::[[Polar coordinates|$4.1.3$]]
:::[[Logarithm (multivalued)|$4.1.4$]]
:::[[Relationship between logarithm (multivalued) and logarithm|$4.1.5$]]
:::[[Logarithm (multivalued) of product is a sum of logarithms (multivalued)|$4.1.6$]]
:::[[Logarithm of product is a sum of logarithms|$4.1.7$]]
:::[[Logarithm (multivalued) of a quotient is a difference of logarithms (multivalued)|$4.1.8$]]
:::[[Logarithm of a quotient is a difference of logarithms|$4.1.9$]]
:::[[Relationship between logarithm (multivalued) and positive integer exponents|$4.1.10$]]
:::[[Relationship between logarithm and positive integer exponents|$4.1.11$]]
:::[[Logarithm of 1|$4.1.12$]]
:::[[Logarithm diverges to negative infinity at 0 from right|$4.1.13$]]
:::[[Logarithm at minus 1|$4.1.14$]]
:::[[Logarithm at i|$4.1.15$]] (also [[Logarithm at -i|$4.1.15$]])
:::[[E|$4.1.16$]]
:::[[E is limit of (1+1/n)^n|$4.1.17$]]
:::$4.2.18$
:::$4.2.19$
:::$4.2.20$
:::$4.2.21$
:::$4.2.22$
:::$4.2.23$
:::[[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 absolute value of (z-1) less than 1|$4.1.26$]]
:::[[Series for log(z) for Re(z) greater than 0|$4.1.27$]]
:::[[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|$4.1.29$]]
::4.2. Exponential Function
:::[[Exponential|$4.2.1$]]
:::[[Logarithm (multivalued) of the exponential|$4.2.2$]]
:::[[Logarithm of exponential|$4.2.3$]]
:::[[Exponential of logarithm|$4.2.4$]]
:::[[Derivative of the exponential function|$4.2.5$]]
:::$4.2.6$
:::$4.2.7$
:::$4.2.8$
:::$4.2.9$
:::$4.2.10$
:::$4.2.11$
:::$4.2.12$
:::$4.2.13$
:::[[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 less than 1/(1-x) for nonzero real x less than 1|$4.2.31$]]
:::[[X/(1+x) less than 1-e^(-x) less than x for nonzero real x greater than -1|$4.2.32$]]
:::[[X less than e^x-1 less than x/(1-x) for nonzero real x less than 1|$4.2.33$]]
:::[[1+x greater than exp(x/(1+x)) for nonzero real x greater than -1|$4.2.34$]]
:::[[E^x greater than 1+x^n/n! for n greater than 0 and nonzero real x greater than 0|$4.2.35$]]
:::[[E^x greater than (1+x/y)^y greater than exp(xy/(x+y) for x greater than 0 and y greater than 0)|$4.2.36$]]
:::[[E^(-x) less than 1-(x/2) for 0 less than x less than or equal to 1.5936|$4.2.37$]]
:::[[Abs(z)/4 less than abs(e^z-1) less than (7abs(z))/4 for 0 less than abs(z) less than 1|$4.2.38$]]
:::[[Abs(e^z-1) less than or equal to e^(abs(z))-1 less than or equal to abs(z)e^(abs(z))|$4.2.39$]]
::4.3. Circular Functions
:::[[Sine|$4.3.1$]]
:::[[Cosine|$4.3.2$]]
:::[[Tangent|$4.3.3$]]
:::[[Cosecant|$4.3.4$]]
:::[[Secant|$4.3.5$]]
:::[[Cotangent|$4.3.6$]]
::4.4. Inverse Circular Functions
::4.5. Hyperbolic Functions
:::[[Sinh|$4.5.1$]]
:::[[Cosh|$4.5.2$]]
:::[[Tanh|$4.5.3$]]
:::[[Csch|$4.5.4$]]
:::[[Sech|$4.5.5$]]
:::[[Coth|$4.5.6$]]
:::[[Relationship between sinh and sin|$4.5.7$]]
:::[[Relationship between cosh and cos|$4.5.8$]]
:::[[Relationship between tanh and tan|$4.5.9$]]
:::[[Relationship between csch and csc|$4.5.10$]]
:::[[Relationship between sech and sec|$4.5.11$]]
:::[[Relationship between coth and cot|$4.5.12$]]
:::[[Period of sinh|$4.5.13$]]
:::[[Period of cosh|$4.5.14$]]
:::[[Period of tanh|$4.5.15$]]
::4.6. Inverse Hyperbolic Functions
::4.7. Use and Extension of the Tables
:5. Exponential Integral and Related Functions
::5.1. Exponential Integral
:::[[Exponential integral E|$5.1.1$]]
:::[[Exponential integral Ei|$5.1.2$]]
:::[[Logarithmic integral|$5.1.3$]]
:::[[Exponential integral E|$5.1.4$]]
:::$5.1.5$
:::$5.1.6$
:::$5.1.7$
:::$5.1.8$
:::$5.1.9$
:::$5.1.10$
:::$5.1.11$
:::$5.1.12$
:::[[Symmetry relation of exponential integral E|$5.1.13$]]
:::[[Recurrence relation of exponential integral E|$5.1.14$]]
::5.2. Sine and Cosine Integrals
:::[[Sine integral|$5.2.1$]]
::5.3. Use and Extension of the Tables
:6. Gamma Function and Related Functions
::6.1. Gamma Function
:::[[Gamma|$6.1.1$]]
:::[[Gauss' formula for gamma function|$6.1.2$]]
:::$6.1.3$
:::$6.1.4$
:::$6.1.5$
:::$6.1.6$
:::$6.1.7$
:::$6.1.8$
:::$6.1.9$
:::$6.1.10$
:::$6.1.11$
:::$6.1.12$
:::$6.1.13$
:::$6.1.14$
:::$6.1.15$
:::$6.1.16$
:::$6.1.17$
:::$6.1.18$
:::$6.1.19$
:::$6.1.20$
:::$6.1.21$
:::$6.1.22$
:::$6.1.23$
:::$6.1.24$
:::$6.1.25$
:::$6.1.26$
:::$6.1.27$
:::$6.1.28$
:::$6.1.29$
:::$6.1.30$
:::$6.1.31$
:::$6.1.32$
:::$6.1.33$
:::$6.1.34$
:::$6.1.35$
:::$6.1.36$
:::$6.1.37$
:::$6.1.38$
:::$6.1.39$
:::$6.1.40$
:::$6.1.41$
:::$6.1.42$
:::$6.1.43$
:::$6.1.44$
:::$6.1.45$
:::$6.1.46$
:::$6.1.47$
:::$6.1.48$
:::$6.1.49$
:::$6.1.50$
::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 in terms of gamma|$6.2.2$]] (and [[Beta is symmetric|$6.2.2$]])
::6.3. Psi (Digamma Function)
:::[[Digamma|$6.3.1$]]
:::[[Digamma at 1|$6.3.2$]] (and [[Digamma at n+1|$6.3.2$]])
:::[[Digamma at 1/2|$6.3.3$]]
:::[[Digamma at n+1/2|$6.3.4$]]
:::[[Digamma functional equation|$6.3.5$]]
::6.4. Polygamma Functions
:::[[Polygamma|$6.4.1$]] (and [[Integral representation of polygamma for Re(z) greater than 0|$6.4.1$]])
:::[[Value of polygamma at 1|$6.4.2$]]
:::[[Value of polygamma at positive integer|$6.4.3$]]
:::[[Value of polygamma at 1/2|$6.4.4$]]
:::[[Value of derivative of trigamma at positive integer plus 1/2|$6.4.5$]]
:::[[Polygamma recurrence relation|$6.4.6$]]
:::[[Polygamma reflection formula|$6.4.7$]]
:::[[Polygamma multiplication formula|$6.4.8$]]
:::$6.4.9$
:::$6.4.10$
:::$6.4.11$
:::$6.4.12$
:::$6.4.13$
:::$6.4.14$
::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
::7.1. Error Function
:::[[Error function|$7.1.1$]]
:::[[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|$7.1.3$]]
:::$7.1.4$
:::[[Taylor series for error function|$7.1.5$]]
:::$7.1.6$
:::$7.1.7$
:::$7.1.8$
:::[[Error function is odd|$7.1.9$]]
:::[[Erf of conjugate is conjugate of erf|$7.1.10$]]
:::$7.1.11$
:::$7.1.12$
:::$7.1.13$
:::[[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|$7.1.15$]]
:::[[Limit of erf when z approaches infinity and the modulus of arg(z) is less than pi/4|$7.1.16$]]
:::$7.1.17$
:::$7.1.18$
:::$7.1.19$
:::$7.1.20$
:::$7.1.21$
:::$7.1.22$
:::$7.1.23$
:::$7.1.24$
:::$7.1.25$
:::$7.1.26$
:::$7.1.27$
:::$7.1.28$
:::$7.1.29$
::7.2. Repeated Integrals of the Error Function
::7.3. Fresnel Integrals
::7.4. Definite and Indefinite Integrals
::7.5. Use and Extension of the Tables
: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
:::$9.1.1$
:::[[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 (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|$9.1.5$]])
:::$9.1.6$
:::$9.1.7$
:::$9.1.8$
:::$9.1.9$
:::[[Bessel J|$9.1.10$]]
:::$9.1.11$
:::$9.1.12$
:::$9.1.13$
:::$9.1.14$
:::$9.1.15$
:::$9.1.16$
:::$9.1.17$
:::$9.1.18$
:::$9.1.19$
:::$9.1.20$
:::$9.1.21$
:::$9.1.22$
:::$9.1.23$
:::$9.1.24$
:::$9.1.25$
:::$9.1.26$
:::$9.1.27$
:::$9.1.28$
:::$9.1.29$
:::$9.1.30$
:::$9.1.31$
:::$9.1.32$
:::$9.1.33$
:::$9.1.34$
:::$9.1.35$
:::$9.1.36$
:::$9.1.37$
:::$9.1.38$
:::$9.1.39$
:::$9.1.40$
:::$9.1.41$
:::$9.1.42$
:::$9.1.43$
:::$9.1.44$
:::$9.1.45$
:::$9.1.46$
:::$9.1.47$
:::$9.1.48$
:::$9.1.49$
:::$9.1.50$
:::$9.1.51$
:::$9.1.52$
:::$9.1.53$
:::$9.1.54$
:::$9.1.55$
:::$9.1.56$
:::$9.1.57$
:::$9.1.58$
:::$9.1.59$
:::$9.1.60$
:::$9.1.61$
:::$9.1.62$
:::[[Derivative of Bessel J with respect to its order|$9.1.64$]]
:::[[Derivative of Bessel Y with respect to its order|$9.1.65$]]
:::$9.1.66$
:::$9.1.67$
:::$9.1.68$
:::$9.1.69$
:::$9.1.70$
:::$9.1.71$
:::$9.1.72$
:::$9.1.73$
:::$9.1.74$
:::$9.1.75$
:::$9.1.76$
:::$9.1.77$
:::$9.1.78$
:::$9.1.79$
:::$9.1.80$
:::$9.1.81$
:::$9.1.82$
:::$9.1.83$
:::$9.1.84$
:::$9.1.85$
:::$9.1.86$
:::$9.1.87$
:::$9.1.88$
:::$9.1.89$
::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
::11.1 Simple Integrals of Bessel Functions
:::[[Integral of monomial times Bessel J|$11.1.1$]]
:::[[Integral of Bessel J for Re(nu) greater than -1|$11.1.2$]]
:::[[Integral of Bessel J for nu=2n|$11.1.3$]]
:::[[Integral of Bessel J for nu=2n+1|$11.1.4$]]
:::[[Integral of Bessel J for nu=n+1|$11.1.5$]]
:::[[Integral of Bessel J for nu=1|$11.1.6$]]
:::$11.1.7$
:::$11.1.8$
:::$11.1.9$
:::$11.1.10$
:::$11.1.11$
:::$11.1.12$
:::$11.1.13$
:::$11.1.14$
:::$11.1.15$
:::$11.1.16$
:::$11.1.17$
:::$11.1.18$
:::$11.1.19$
:::$11.1.20$
:::$11.1.21$
:::$11.1.22$
:::$11.1.23$
:::$11.1.24$
:::$11.1.25$
:::$11.1.26$
:::$11.1.27$
:::$11.1.28$
:::$11.1.29$
:::$11.1.30$
:::$11.1.31$
::11.2 Repeated Integrals of $J_n(z)$ and $K_0(z)$
::11.3 Reduction Formulas for Indefinite Integrals
::11.4 Definite Integrals
::11.5 Use and Extensions of the Tables
: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
:::[[Anger function|$12.3.1$]]<br />
:::[[Anger of integer order is Bessel J|$12.3.2$]]<br />
:::[[Weber function|$12.3.3$]]<br />
:::[[Relationship between Anger function and Weber function|$12.3.4$]]<br />
:::[[Relationship between Weber function and Anger function|$12.3.5$]]<br />
:::$12.3.6$
:::$12.3.7$
:::$12.3.8$
:::$12.3.9$
:::$12.3.10$
::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
:::[[Hypergeometric 2F1|$15.1.1$]]
:::[[Limit of (1/Gamma(c))*2F1(a,b;c;z) as c approaches -m|$15.1.2$]]
:::[[2F1(1,1;2;z)=-log(1-z)/z|$15.1.3$]]
:::[[2F1(1/2,1;3/2;z^2)=log((1+z)/(1-z))/(2z)|$15.1.4$]]
:::[[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|$15.1.6$]]
:::$15.1.7$
:::$15.1.8$
:::$15.1.9$
:::$15.1.10$
:::$15.1.11$
:::$15.1.12$
:::$15.1.13$
:::$15.1.14$
:::$15.1.15$
:::$15.1.16$
:::$15.1.17$
:::$15.1.18$
:::$15.1.19$
:::$15.1.20$
:::$15.1.21$
:::$15.1.22$
:::$15.1.23$
:::$15.1.24$
:::$15.1.25$
:::$15.1.26$
:::$15.1.27$
:::$15.1.28$
:::$15.1.29$
:::$15.1.30$
:::$15.1.31$
::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 $q$
:::[[Jacobi theta 1|$16.27.1$]]
:::[[Jacobi theta 2|$16.27.2$]]
:::[[Jacobi theta 3|$16.27.3$]]
:::[[Jacobi theta 4|$16.27.4$]]
::16.28 Relations Between the Squares of the Theta Functions
:::[[Squares of theta relation for Jacobi theta 1 and Jacobi theta 4|$16.28.1$]]
:::[[Squares of theta relation for Jacobi theta 2 and Jacobi theta 4|$16.28.2$]]
:::[[Squares of theta relation for Jacobi theta 3 and Jacobi theta 4|$16.28.3$]]
:::[[Squares of theta relation for Jacobi theta 4 and Jacobi theta 4|$16.28.4$]]
:::[[Sum of fourth powers of Jacobi theta 2 and Jacobi theta 4 equals fourth power of Jacobi theta 3|$16.28.5$]]
:::[[Derivative of Jacobi theta 1 at 0|$16.28.6$]]
:::[[Sum of fourth powers of Jacobi theta 2 and Jacobi theta 4 equals fourth power of Jacobi theta 3|$16.28.5$]]
:::[[Derivative of Jacobi theta 1 at 0|$16.28.6$]]
::16.29 Logarithmic Derivatives of the Theta Functions
:::[[Logarithmic derivative of Jacobi theta 1 equals cotangent + a sum of sines|$16.29.1$]]
:::[[Logarithmic derivative of Jacobi theta 2 equals negative tangent + a sum of sines|$16.29.2$]]
:::[[Logarithmic derivative of Jacobi theta 3 equals a sum of sines|$16.29.3$]]
:::[[Logarithmic derivative of Jacobi theta 4 equals a sum of sines|$16.29.4$]]
::16.30 Logarithms of Theta Functions of Sum and Difference
:::[[Logarithm of quotient of Jacobi theta 1 equals the log of a quotient of sines + a sum of sines|$16.30.1$]]
:::[[Logarithm of quotient of Jacobi theta 2 equals the log of a quotient of cosines + a sum of sines|$16.30.2$]]
:::[[Logarithm of quotient of Jacobi theta 3 equals a sum of sines|$16.30.3$]]
:::[[Logarithm of a quotient of Jacobi theta 4 equals a sum of sines|$16.30.4$]]
::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
::23.1 Bernoulli and Euler Polynomials and the Euler-Maclaurin formula
::23.2 Riemann Zeta Function and Other Sums of Reciprocal Powers
:::[[Riemann zeta|$23.2.1$]]
:::[[Euler product for Riemann zeta|$23.2.2$]]
: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
::::[[Partition|$24.2.1 \mathrm{I}.A.$]]
::::[[Generating function for partition function|$24.2.1 \mathrm{I}.B.$]]
::::[[Closed form for partition function with sinh|$24.2.1 \mathrm{II}.A.$]]
::::[[Pure recurrence relation for partition function|$24.2.1 \mathrm{II}.A.$]]
::::[[Recurrence relation for partition function with sum of divisors|$24.2.1 \mathrm{II}.A.$]]
::::[[Sum of divisors functions written in terms of partition function|$24.2.1 \mathrm{II}.B.$]]
::::[[Asymptotic formula for partition function|$24.2.1 \mathrm{III}$]]
:::24.2.2. Partitions Into Distinct Parts
::24.3. Number Theoretic Functions
:::24.3.1. The Mobius Function
::::[[Möbius function|$24.3.1 \mathrm{I}.A.$]]
::::[[Reciprocal of Riemann zeta as a sum of Möbius function for Re(z) greater than 1|$24.3.1. \mathrm{I}.B.$]]
::::[[Identity written as a sum of Möbius functions|$24.3.1 \mathrm{I}.B.$]]
::::[[Möbius function is multiplicative|$24.3.1 \mathrm{II}.A.$]]
:::24.3.2. The Euler Function
::::[[Euler totient|$24.3.2. \mathrm{I}.A$]]
::::[[Sum of totient equals zeta(z-1)/zeta(z) for Re(z) greater than 2|$24.3.2 \mathrm{I}.B.$]]
::::[[Sum of totient equals z/((1-z) squared)|$24.3.2 \mathrm{I}.B.$]]
::::[[Product representation of totient|$24.3.2 \mathrm{I}.C.$]]
::::[[Euler totient is multiplicative|$24.3.2 \mathrm{II}.A.$]]
:::24.3.3. Divisor Functions
::::[[Sum of divisors|$24.3.3I.A.$]]
::::[[Sum of sum of divisors function equals product of Riemann zeta for Re(z) greater than k+1|$24.3.3I.B.$]]
:::24.3.4. Primitive Roots
:::References
:25. Numerical Interpolation, Differentiation and Integration
:26. Probability Functions
:27. Miscellaneous Functions
:28. Scales of Notation
:29. Laplace Transforms

:Subject Index
:Index of Notations

[[Category:Book]]