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

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===Online mirrors===
 
===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://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 />
−
[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 />
 
[http://www.math.hkbu.edu.hk/support/aands/ Hong Kong Baptist University]<br />
  
Line 68: Line 66:
 
:::[[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|$4.1.25$]]
Line 80: Line 78:
 
:::[[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|$4.2.1$]]
−
:::[[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
 
::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
 
::4.4. Inverse Circular Functions
 
::4.5. Hyperbolic Functions
 
::4.5. Hyperbolic Functions
−
:::[[Sinh|4.5.1]]
+
:::[[Sinh|$4.5.1$]]
−
:::[[Cosh|4.5.2]]
+
:::[[Cosh|$4.5.2$]]
−
:::[[Tanh|4.5.3]]
+
:::[[Tanh|$4.5.3$]]
−
:::[[Csch|4.5.4]]
+
:::[[Csch|$4.5.4$]]
−
:::[[Sech|4.5.5]]
+
:::[[Sech|$4.5.5$]]
−
:::[[Coth|4.5.6]]
+
:::[[Coth|$4.5.6$]]
−
:::[[Relationship between sinh and sin|4.5.7]]
+
:::[[Relationship between sinh and sin|$4.5.7$]]
−
:::[[Relationship between cosh and cos|4.5.8]]
+
:::[[Relationship between cosh and cos|$4.5.8$]]
−
:::[[Relationship between tanh and tan|4.5.9]]
+
:::[[Relationship between tanh and tan|$4.5.9$]]
−
:::[[Relationship between csch and csc|4.5.10]]
+
:::[[Relationship between csch and csc|$4.5.10$]]
−
:::[[Relationship between sech and sec|4.5.11]]
+
:::[[Relationship between sech and sec|$4.5.11$]]
−
:::[[Relationship between coth and cot|4.5.12]]
+
:::[[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
 
::4.6. Inverse Hyperbolic Functions
 
::4.7. Use and Extension of the Tables
 
::4.7. Use and Extension of the Tables
 
:5. Exponential Integral and Related Functions
 
:5. Exponential Integral and Related Functions
 
::5.1. Exponential Integral
 
::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
 
::5.2. Sine and Cosine Integrals
 +
:::[[Sine integral|$5.2.1$]]
 
::5.3. Use and Extension of the Tables
 
::5.3. Use and Extension of the Tables
 
:6. Gamma Function and Related Functions
 
:6. Gamma Function and Related Functions
 
::6.1. Gamma Function
 
::6.1. Gamma Function
−
:::[[Gamma|6.1.1]]
+
:::[[Gamma|$6.1.1$]]
−
:::[[Gauss' formula for gamma function|6.1.2]]
+
:::[[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
 
::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)
 
::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
 
::6.4. Polygamma Functions
−
:::[[Polygamma|6.4.1]] (and [[Integral representation of polygamma for Re(z) greater than 0|6.4.1]])
+
:::[[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 1|$6.4.2$]]
−
:::[[Value of polygamma at positive integer|6.4.3]]
+
:::[[Value of polygamma at positive integer|$6.4.3$]]
−
:::[[Value of polygamma at 1/2|6.4.4]]
+
:::[[Value of polygamma at 1/2|$6.4.4$]]
−
:::[[Value of derivative of trigamma at positive integer plus 1/2|6.4.5]]
+
:::[[Value of derivative of trigamma at positive integer plus 1/2|$6.4.5$]]
−
:::[[Polygamma recurrence relation|6.4.6]]
+
:::[[Polygamma recurrence relation|$6.4.6$]]
−
:::[[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|$6.4.9$]]
−
:::6.4.10
 
−
:::6.4.11
 
−
:::6.4.12
 
−
:::6.4.13
 
−
:::6.4.14
 
 
::6.5. Incomplete Gamma Function
 
::6.5. Incomplete Gamma Function
 
::6.6. Incomplete Beta Function
 
::6.6. Incomplete Beta Function
Line 208: Line 209:
 
:7. Error Function and Fresnel Integrals
 
:7. Error Function and Fresnel Integrals
 
::7.1. Error Function
 
::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
 
::7.2. Repeated Integrals of the Error Function
 
::7.3. Fresnel Integrals
 
::7.3. Fresnel Integrals
Line 259: Line 243:
 
:9. Bessel Functions of Integer Order
 
:9. Bessel Functions of Integer Order
 
::9.1. Definitions and Elementary Properties
 
::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|$9.1.5$]])
−
:::9.1.6
+
:::-----------
−
:::9.1.7
+
:::[[Bessel J|$9.1.10$]]
−
:::9.1.8
+
:::-----------
−
:::9.1.9
+
:::[[Derivative of Bessel J with respect to its order|$9.1.64$]]
−
:::[[Bessel J|9.1.10]]
+
:::[[Derivative of Bessel Y with respect to its order|$9.1.65$]]
−
:::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
 
::9.2. Asymptotic Expansions for Large Arguments
 
::9.3. Asymptotic Expansions for Large Orders
 
::9.3. Asymptotic Expansions for Large Orders
Line 372: 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|$11.1.6$]]
−
:::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 $K_0(z)$
 
::11.2 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 403: 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
−
:::[[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
 
::12.4 Use and Extension of the Tables
 
:13. Confluent Hypergeometric Functions
 
:13. Confluent Hypergeometric Functions
Line 437: Line 329:
 
:15. Hypergeometric Functions
 
:15. Hypergeometric Functions
 
::15.1 Gauss Series, Special Elementary Cases, Special Values of the Argument
 
::15.1 Gauss Series, Special Elementary Cases, Special Values of the Argument
−
:::[[Hypergeometric 2F1|15.1.1]]
+
:::[[Hypergeometric 2F1|$15.1.1$]]
−
:::[[Limit of (1/Gamma(c))*2F1(a,b;c;z) as c approaches -m|15.1.2]]
+
:::[[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(1,1;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)=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;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|$15.1.6$]]
−
:::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
 
::15.2 Differentiation Formulas and Gauss' Relations for Contiguous Functions
 
::15.3 Integral Representations and Transformation Formulas
 
::15.3 Integral Representations and Transformation Formulas
Line 502: Line 369:
 
::16.26 Integrals in Terms of the Elliptic Integral of the Second Kind
 
::16.26 Integrals in Terms of the Elliptic Integral of the Second Kind
 
::16.27 Theta Functions; Expansions in Terms of the Nome $q$
 
::16.27 Theta Functions; Expansions in Terms of the Nome $q$
−
:::[[Jacobi theta 1|16.27.1]]
+
:::[[Jacobi theta 1|$16.27.1$]]
−
:::[[Jacobi theta 2|16.27.2]]
+
:::[[Jacobi theta 2|$16.27.2$]]
−
:::[[Jacobi theta 3|16.27.3]]
+
:::[[Jacobi theta 3|$16.27.3$]]
−
:::[[Jacobi theta 4|16.27.4]]
+
:::[[Jacobi theta 4|$16.27.4$]]
 
::16.28 Relations Between the Squares of the Theta Functions
 
::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 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 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 3 and Jacobi theta 4|$16.28.3$]]
−
:::[[Squares of theta relation for Jacobi theta 4 and Jacobi theta 4|16.28.4]]
+
:::[[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]]
+
:::[[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]]
+
:::[[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]]
+
:::[[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]]
+
:::[[Derivative of Jacobi theta 1 at 0|$16.28.6$]]
 
::16.29 Logarithmic Derivatives of the Theta Functions
 
::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 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 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 3 equals a sum of sines|$16.29.3$]]
−
:::[[Logarithmic derivative of Jacobi theta 4 equals a sum of sines|16.29.4]]
+
:::[[Logarithmic derivative of Jacobi theta 4 equals a sum of sines|$16.29.4$]]
 
::16.30 Logarithms of Theta Functions of Sum and Difference
 
::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 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 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 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]]
+
:::[[Logarithm of a quotient of Jacobi theta 4 equals a sum of sines|$16.30.4$]]
 
::16.31 Jacobi's Notation for Theta Functions
 
::16.31 Jacobi's Notation for Theta Functions
 
::16.32 Calculation of Jacobi's Theta Function $\Theta(u|m)$ by Use of the Arithmetic-Geometric Mean
 
::16.32 Calculation of Jacobi's Theta Function $\Theta(u|m)$ by Use of the Arithmetic-Geometric Mean
Line 538: Line 405:
 
::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
Line 565: Line 436:
 
:22. Orthogonal Polynomials
 
:22. Orthogonal Polynomials
 
:23. Bernoulli and Euler Polynomials, Riemann Zeta Function
 
: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. Combinatorial Analysis
 
::24.1. Basic numbers
 
::24.1. Basic numbers
Line 601: Line 476:
 
: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
$3.3.1$
$3.3.2$
$3.3.3$
$3.3.4$
$3.3.5$
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
$4.1.1$
$4.1.2$
$4.1.3$
$4.1.4$
$4.1.5$
$4.1.6$
$4.1.7$
$4.1.8$
$4.1.9$
$4.1.10$
$4.1.11$
$4.1.12$
$4.1.13$
$4.1.14$
$4.1.15$ (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
$4.2.1$
$4.2.2$
$4.2.3$
$4.2.4$
$4.2.5$
----------
$4.2.29$
$4.2.30$
$4.2.31$
$4.2.32$
$4.2.33$
$4.2.34$
$4.2.35$
$4.2.36$
$4.2.37$
$4.2.38$
$4.2.39$
4.3. Circular Functions
$4.3.1$
$4.3.2$
$4.3.3$
$4.3.4$
$4.3.5$
$4.3.6$
- - - - -
$4.3.105$
$4.3.106$
$4.3.107$
$4.3.108$
$4.3.109$
$4.3.110$
- - - - -
$4.3.147$ (or $4.3.147$ or $4.3.147$ or $4.3.147$)
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
5. Exponential Integral and Related Functions
5.1. Exponential Integral
$5.1.1$
$5.1.2$
$5.1.3$
$5.1.4$
----------
$5.1.13$
$5.1.14$
5.2. Sine and Cosine Integrals
$5.2.1$
5.3. Use and Extension of the Tables
6. Gamma Function and Related Functions
6.1. Gamma Function
$6.1.1$
$6.1.2$
---------------
6.2. Beta Function
$6.2.1$ (and $6.2.1$ and $6.2.1$)
$6.2.2$ (and $6.2.2$)
6.3. Psi (Digamma Function)
$6.3.1$
$6.3.2$ (and $6.3.2$)
$6.3.3$
$6.3.4$
$6.3.5$
6.4. Polygamma Functions
$6.4.1$ (and $6.4.1$)
$6.4.2$
$6.4.3$
$6.4.4$
$6.4.5$
$6.4.6$
$6.4.7$
$6.4.8$
$6.4.9$
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
$7.1.1$
$7.1.2$
$7.1.3$
-------------
$7.1.5$
----------
$7.1.9$
$7.1.10$
-----------
$7.1.14$
$7.1.15$
$7.1.16$
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
-----------
$9.1.2$
$9.1.3$ (and $9.1.3$)
$9.1.4$ (and $9.1.4$)
$9.1.5$ (and $9.1.5$)
-----------
$9.1.10$
-----------
$9.1.64$
$9.1.65$
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
$11.1.1$
$11.1.2$
$11.1.3$
$11.1.4$
$11.1.5$
$11.1.6$
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 $\mathbf{H}_{\nu}(z)$
--------------------
$12.1.3$
--------------------
$12.1.6$
$12.1.7$
$12.1.8$
$12.1.9$
$12.1.10$
$12.1.11$
$12.1.12$
$12.1.13$
$12.1.14$
$12.1.15$
$12.1.16$
$12.1.17$
12.2 Modified Struve Functions $\mathbf{L}_{\nu}(z)$
12.3 Anger and Weber Functions
$12.3.1$
$12.3.2$
$12.3.3$
$12.3.4$
$12.3.5$
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
$15.1.1$
$15.1.2$
$15.1.3$
$15.1.4$
$15.1.5$
$15.1.6$
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 $q$
$16.27.1$
$16.27.2$
$16.27.3$
$16.27.4$
16.28 Relations Between the Squares of the Theta Functions
$16.28.1$
$16.28.2$
$16.28.3$
$16.28.4$
$16.28.5$
$16.28.6$
$16.28.5$
$16.28.6$
16.29 Logarithmic Derivatives of the Theta Functions
$16.29.1$
$16.29.2$
$16.29.3$
$16.29.4$
16.30 Logarithms of Theta Functions of Sum and Difference
$16.30.1$
$16.30.2$
$16.30.3$
$16.30.4$
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
$17.3.1$
----------
$17.3.9$
$17.3.10$
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
$23.2.1$
$23.2.2$
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
$24.2.1 \mathrm{I}.A.$
$24.2.1 \mathrm{I}.B.$
$24.2.1 \mathrm{II}.A.$
$24.2.1 \mathrm{II}.A.$
$24.2.1 \mathrm{II}.A.$
$24.2.1 \mathrm{II}.B.$
$24.2.1 \mathrm{III}$
24.2.2. Partitions Into Distinct Parts
24.3. Number Theoretic Functions
24.3.1. The Mobius Function
$24.3.1 \mathrm{I}.A.$
$24.3.1. \mathrm{I}.B.$
$24.3.1 \mathrm{I}.B.$
$24.3.1 \mathrm{II}.A.$
24.3.2. The Euler Function
$24.3.2. \mathrm{I}.A$
$24.3.2 \mathrm{I}.B.$
$24.3.2 \mathrm{I}.B.$
$24.3.2 \mathrm{I}.C.$
$24.3.2 \mathrm{II}.A.$
24.3.3. Divisor Functions
$24.3.3I.A.$
$24.3.3I.B.$
24.3.4. Primitive Roots
References
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
$29.1.1$
Subject Index
Index of Notations
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