Difference between revisions of "Book:Milton Abramowitz/Handbook of mathematical functions"

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},
}

Difference between revisions of "Book:Milton Abramowitz/Handbook of mathematical functions"

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

Online mirrors

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@@ Line 4: / Line 4: @@
 ===Online mirrors===
-[http://specialfunctionswiki.org/mirror/abramowitz_and_stegun-1.03/ Hosted by specialfunctionswiki]<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://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 />
@@ Line 21: / Line 19: @@
        YEAR = {1964},
       PAGES = {xiv+1046},
-   MRCLASS = {33.00 (65.05)},
-  MRNUMBER = {0167642},
-MRREVIEWER = {D. H. Lehmer},
 }</pre>
@@ Line 38: / Line 33: @@
 ::3.2. Inequalities
 ::3.3. Rules for Differentiation and Integration
-:::[[Constant multiple rule for derivatives|3.3.1]]
+:::[[Constant multiple rule for derivatives|$3.3.1$]]
-:::[[Sum rule for derivatives|3.3.2]]
+:::[[Sum rule for derivatives|$3.3.2$]]
-:::[[Product rule for derivatives|3.3.3]]
+:::[[Product rule for derivatives|$3.3.3$]]
-:::[[Quotient rule for derivatives|3.3.4]]
+:::[[Quotient rule for derivatives|$3.3.4$]]
-:::[[Chain rule for derivatives|3.3.5]]
+:::[[Chain rule for derivatives|$3.3.5$]]
 ::3.4. Limits, Maxima and Minima
 ::3.5. Absolute and Relative Errors
@@ Line 54: / Line 49: @@
 :4. Elementary Transcendental Functions: Logarithmic, Exponential, Circular and Hyperbolic Functions
 ::4.1. Logarithmic Function
-:::[[Logarithm|4.1.1]]
+:::[[Logarithm|$4.1.1$]]
-:::[[Real and imaginary parts of log|4.1.2]]
+:::[[Real and imaginary parts of log|$4.1.2$]]
-:::[[Polar coordinates|4.1.3]]
+:::[[Polar coordinates|$4.1.3$]]
-:::[[Logarithm (multivalued)|4.1.4]]
+:::[[Logarithm (multivalued)|$4.1.4$]]
-:::[[Relationship between logarithm (multivalued) and logarithm|4.1.5]]
+:::[[Relationship between logarithm (multivalued) and logarithm|$4.1.5$]]
-:::[[Logarithm (multivalued) of product is a sum of logarithms (multivalued)|4.1.6]]
+:::[[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 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 (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]]
+:::[[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 (multivalued) and positive integer exponents|$4.1.10$]]
-:::[[Relationship between logarithm and positive integer exponents|4.1.11]]
+:::[[Relationship between logarithm and positive integer exponents|$4.1.11$]]
-:::[[Logarithm of 1|4.1.12]]
+:::[[Logarithm of 1|$4.1.12$]]
-:::[[Logarithm diverges to negative infinity at 0 from right|4.1.13]]
+:::[[Logarithm diverges to negative infinity at 0 from right|$4.1.13$]]
-:::[[Logarithm at minus 1|4.1.14]]
+:::[[Logarithm at minus 1|$4.1.14$]]
-:::[[Logarithm at i|4.1.15]] (also [[Logarithm at -i|4.1.15]])
+:::[[Logarithm at i|$4.1.15$]] (also [[Logarithm at -i|$4.1.15$]])
-:::[[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|$4.1.17$]]
-:::4.2.18
+:::[[Logarithm base a|$4.1.18$]]
-:::4.2.19
+:::[[Log base a in terms of logarithm base b|$4.1.19$]]
-:::4.2.20
+:::[[Log a(z)=1/log b(a)|$4.1.20$]]
-:::4.2.21
+:::[[Log e(z)=log(z)|$4.1.21$]]
-:::4.2.22
+:::[[Log 10(z)=log(z)/log(10)|$4.1.22$]] (and [[Log 10(z)=log 10(e)log(z)|$4.1.22$]])
-:::4.2.23
+:::[[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|$4.1.25$]]
-:::[[Series for log(z) for absolute value of (z-1) less than 1|4.1.26]]
+:::[[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]]
+:::[[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]]
+:::[[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
-:::[[Exponential|4.2.1]]
+:::[[Exponential|$4.2.1$]]
-:::[[Logarithm (multivalued) of the exponential|4.2.2]]
+:::[[Logarithm (multivalued) of the exponential|$4.2.2$]]
-:::[[Logarithm of exponential|4.2.3]]
+:::[[Logarithm of exponential|$4.2.3$]]
-:::[[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|$4.2.5$]]
-:::4.2.6
+:::----------
-:::4.2.7
+:::[[E^(-x/(1-x)) is less than 1-x is less than e^(-x) for nonzero real x less than 1|$4.2.29$]]
-:::4.2.8
+:::[[E^x is greater than 1+x for nonzero real x|$4.2.30$]]
-:::4.2.9
+:::[[E^x is less than 1/(1-x) for nonzero real x less than 1|$4.2.31$]]
-:::4.2.10
+:::[[X/(1+x) less than 1-e^(-x) less than x for nonzero real x greater than -1|$4.2.32$]]
-:::4.2.11
+:::[[X less than e^x-1 less than x/(1-x) for nonzero real x less than 1|$4.2.33$]]
-:::4.2.12
+:::[[1+x greater than exp(x/(1+x)) for nonzero real x greater than -1|$4.2.34$]]
-:::4.2.13
+:::[[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/(1-x)) is less than 1-x is less than e^(-x) for nonzero real x less than 1|4.2.29]]
+:::[[E^(-x) less than 1-(x/2) for 0 less than x less than or equal to 1.5936|$4.2.37$]]
-:::[[E^x is greater than 1+x for nonzero real x|4.2.30]]
+:::[[Abs(z)/4 less than abs(e^z-1) less than (7abs(z))/4 for 0 less than abs(z) less than 1|$4.2.38$]]
-:::[[E^x is less than 1/(1-x) for nonzero real x less than 1|4.2.31]]
+:::[[Abs(e^z-1) less than or equal to e^(abs(z))-1 less than or equal to abs(z)e^(abs(z))|$4.2.39$]]
-:::[[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]]
+:::[[Sine|$4.3.1$]]
-:::[[Cosine|4.3.2]]
+:::[[Cosine|$4.3.2$]]
-:::[[Tangent|4.3.3]]
+:::[[Tangent|$4.3.3$]]
-:::[[Cosecant|4.3.4]]
+:::[[Cosecant|$4.3.4$]]
-:::[[Secant|4.3.5]]
+:::[[Secant|$4.3.5$]]
-:::[[Cotangent|4.3.6]]
+:::[[Cotangent|$4.3.6$]]
+::: - - - - -
+:::
+:::[[Derivative of sine|$4.3.105$]]
+:::[[Derivative of cosine|$4.3.106$]]
+:::[[Derivative of tangent|$4.3.107$]]
+:::[[Derivative of cosecant|$4.3.108$]]
+:::[[Derivative of secant|$4.3.109$]]
+:::[[Derivative of cotangent|$4.3.110$]]
+::: - - - - -
+:::
+::: [[Versine|$4.3.147$]] (or [[Coversine|$4.3.147$]] or [[Haversine|$4.3.147$]] or [[Exsecant|$4.3.147$]])
 ::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|$4.5.15$]]
+:::[[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
 ::4.7. 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|$5.1.4$]]
+:::----------
+:::[[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|$6.1.2$]]
+:::---------------
 ::6.2. 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|$6.4.8$]]
+:::[[Series for polygamma in terms of Riemann zeta|$6.4.9$]]
 ::6.5. Incomplete Gamma Function
 ::6.6. Incomplete Beta Function
@@ Line 135: / Line 209: @@
 :7. Error Function and Fresnel Integrals
 ::7.1. Error Function
-:::[[Error function|7.1.1]]
+:::[[Error function|$7.1.1$]]
-:::[[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|$7.1.3$]]
-:::7.1.4
+:::-------------
-:::[[Taylor series for error function|7.1.5]]
+:::[[Taylor series for error function|$7.1.5$]]
-:::7.1.6
+:::----------
-:::7.1.7
+:::[[Error function is odd|$7.1.9$]]
-:::7.1.8
+:::[[Erf of conjugate is conjugate of erf|$7.1.10$]]
-:::[[Error function is odd|7.1.9]]
+:::-----------
-:::[[Erf of conjugate is conjugate of erf|7.1.10]]
+:::[[Continued fraction for 2e^(z^2) integral from z to infinity e^(-t^2) dt for positive Re(z)|$7.1.14$]]
-:::7.1.11
+:::[[Continued fraction for 1/sqrt(pi) integral from -infinity to infinity of e^(-t^2)/(z-t) dt|$7.1.15$]]
-:::7.1.12
+:::[[Limit of erf when z approaches infinity and the modulus of arg(z) is less than pi/4|$7.1.16$]]
-:::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
@@ Line 186: / Line 243: @@
 :9. Bessel Functions of Integer Order
 ::9.1. Definitions and Elementary Properties
-:::9.1.1
+:::-----------
-:::[[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|$9.1.5$]])
+:::-----------
+:::[[Bessel J|$9.1.10$]]
+:::-----------
+:::[[Derivative of Bessel J with respect to its order|$9.1.64$]]
+:::[[Derivative of Bessel Y with respect to its order|$9.1.65$]]
 ::9.2. Asymptotic Expansions for Large Arguments
 ::9.3. Asymptotic Expansions for Large Orders
@@ Line 210: / Line 272: @@
 :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|$11.1.6$]]
 ::11.2 Repeated Integrals of $J_n(z)$ and $K_0(z)$
 ::11.3 Reduction Formulas for Indefinite Integrals
@@ Line 216: / Line 284: @@
 :12. Struve Functions and Related Functions
 ::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 $\mathbf{L}_{\nu}(z)$
 ::12.3 Anger and Weber Functions
-:::[[Anger function|12.3.1]]<br />
+:::[[Anger function|$12.3.1$]]<br />
-:::[[Anger of integer order is Bessel J|12.3.2]]<br />
+:::[[Anger of integer order is Bessel J|$12.3.2$]]<br />
-:::[[Weber function|12.3.3]]<br />
+:::[[Weber function|$12.3.3$]]<br />
-:::[[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 />
-:::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
@@ Line 250: / Line 329: @@
 :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|$15.1.6$]]
 ::15.2 Differentiation Formulas and Gauss' Relations for Contiguous Functions
 ::15.3 Integral Representations and Transformation Formulas
@@ Line 284: / Line 369: @@
 ::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
@@ Line 300: / Line 405: @@
 ::17.2 Canonical Forms
 ::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
 ::17.5 Landen's Transformation
@@ Line 327: / Line 436: @@
 :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
+::-------
+::[[Dilogarithm|$27.7.2$]]
 :28. Scales of Notation
 :29. Laplace Transforms
+::29.1 Definition of the Laplace Transform
+:::[[Laplace transform|$29.1.1$]]
 :Subject Index
 :Index of Notations
-[[Category:Books]]
+[[Category:Book]]