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1: 22.17 Moduli Outside the Interval [0,1]
§22.17 Moduli Outside the Interval [0,1]
k 1 = k 1 + k 2 ,
k 1 k 1 = k 1 + k 2 ,
22.17.7 cn ( z , i k ) = cd ( z / k 1 , k 1 ) ,
For proofs of these results and further information see Walker (2003).
2: 22.7 Landen Transformations
22.7.1 k 1 = 1 - k 1 + k ,
k 2 = 2 k 1 + k ,
k 2 = 1 - k 1 + k ,
22.7.6 sn ( z , k ) = ( 1 + k 2 ) sn ( z / ( 1 + k 2 ) , k 2 ) cn ( z / ( 1 + k 2 ) , k 2 ) dn ( z / ( 1 + k 2 ) , k 2 ) ,
22.7.8 dn ( z , k ) = ( 1 - k 2 ) ( dn 2 ( z / ( 1 + k 2 ) , k 2 ) + k 2 ) k 2 2 dn ( z / ( 1 + k 2 ) , k 2 ) .
3: Sidebar 5.SB1: Gamma & Digamma Phase Plots
In the upper half of the image, the poles of Γ ( z ) are clearly visible at negative integer values of z : the phase changes by 2 π around each pole, showing a full revolution of the color wheel. … Phase changes around the zeros are of opposite sign to those around the poles. …
4: 19.7 Connection Formulas
§19.7(ii) Change of Modulus and Amplitude
κ = k 1 + k 2 ,
κ = 1 1 + k 2 ,
With sinh ϕ = tan ψ , …
§19.7(iii) Change of Parameter of Π ( ϕ , α 2 , k )
5: 19.8 Quadratic Transformations
k 1 = 1 - k 1 + k ,
c 1 = csc 2 ϕ 1 .
k 2 = 2 k / ( 1 + k ) ,
k 1 = ( 1 - k ) / ( 1 + k ) ,
c = csc 2 ϕ .
6: 7.16 Generalized Error Functions
These functions can be expressed in terms of the incomplete gamma function γ ( a , z ) 8.2(i)) by change of integration variable.
7: 27 Functions of Number Theory
8: Foreword
Much has changed in the years since A&S was published. …However, we have also seen the birth of a new age of computing technology, which has not only changed how we utilize special functions, but also how we communicate technical information. …
9: 1.13 Differential Equations
§1.13(iv) Change of Variables
Transformation of the Point at Infinity
Elimination of First Derivative by Change of Dependent Variable
Elimination of First Derivative by Change of Independent Variable
Liouville Transformation
10: 23.21 Physical Applications
Another form is obtained by identifying e 1 , e 2 , e 3 as lattice roots (§23.3(i)), and setting …
23.21.5 ( ( v ) - ( w ) ) ( ( w ) - ( u ) ) ( ( u ) - ( v ) ) 2 = ( ( w ) - ( v ) ) 2 u 2 + ( ( u ) - ( w ) ) 2 v 2 + ( ( v ) - ( u ) ) 2 w 2 .