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1: 19.3 Graphics
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Figure 19.3.7: K ( k ) as a function of complex k 2 for 2 ( k 2 ) 2 , 2 ( k 2 ) 2 . … Magnify 3D Help
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Figure 19.3.8: E ( k ) as a function of complex k 2 for 2 ( k 2 ) 2 , 2 ( k 2 ) 2 . … Magnify 3D Help
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Figure 19.3.9: ( K ( k ) ) as a function of complex k 2 for 2 ( k 2 ) 2 , 2 ( k 2 ) 2 . … Magnify 3D Help
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Figure 19.3.10: ( K ( k ) ) as a function of complex k 2 for 2 ( k 2 ) 2 , 2 ( k 2 ) 2 . … Magnify 3D Help
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Figure 19.3.12: ( E ( k ) ) as a function of complex k 2 for 2 ( k 2 ) 2 , 2 ( k 2 ) 2 . … Magnify 3D Help
2: 29.16 Asymptotic Expansions
The approximations for Lamé polynomials hold uniformly on the rectangle 0 z K , 0 z K , when n k and n k assume large real values. …
3: 7.9 Continued Fractions
7.9.3 w ( z ) = i π 1 z 1 2 z 1 z 3 2 z 2 z , z > 0 .
4: 28.25 Asymptotic Expansions for Large z
28.25.4 z + , π + δ ph h + z 2 π δ ,
28.25.5 z + , 2 π + δ ph h + z π δ ,
5: 19.32 Conformal Map onto a Rectangle
19.32.2 d z = 1 2 ( j = 1 3 ( p x j ) 1 / 2 ) d p , p > 0 ; 0 < ph ( p x j ) < π , j = 1 , 2 , 3 .
6: 23.1 Special Notation
𝕃 lattice in .
z = x + i y complex variable, except in §§23.20(ii), 23.21(iii).
2 ω 1 , 2 ω 3 lattice generators ( ( ω 3 / ω 1 ) > 0 ).
τ = ω 3 / ω 1 lattice parameter ( τ > 0 ).
Whittaker and Watson (1927) requires only ( ω 3 / ω 1 ) 0 , instead of ( ω 3 / ω 1 ) > 0 . …
7: 27.14 Unrestricted Partitions
27.14.12 η ( τ ) = e π i τ / 12 n = 1 ( 1 e 2 π i n τ ) , τ > 0 .
27.14.14 η ( a τ + b c τ + d ) = ε ( i ( c τ + d ) ) 1 2 η ( τ ) ,
27.14.16 Δ ( τ ) = ( 2 π ) 12 ( η ( τ ) ) 24 , τ > 0 ,
27.14.17 Δ ( a τ + b c τ + d ) = ( c τ + d ) 12 Δ ( τ ) ,
8: 20.14 Methods of Computation
For values of | q | near 1 the transformations of §20.7(viii) can be used to replace τ with a value that has a larger imaginary part and hence a smaller value of | q | . …In theory, starting from any value of τ , a finite number of applications of the transformations τ τ + 1 and τ 1 / τ will result in a value of τ with τ 3 / 2 ; see §23.18. In practice a value with, say, τ 1 / 2 , | q | 0.2 , is found quickly and is satisfactory for numerical evaluation.
9: 23.11 Integral Representations
provided that 1 < ( z + τ ) < 1 and | z | < τ .
10: 5.4 Special Values and Extrema
5.4.16 ψ ( i y ) = 1 2 y + π 2 coth ( π y ) ,
5.4.18 ψ ( 1 + i y ) = 1 2 y + π 2 coth ( π y ) .