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11: 21.4 Graphics
Figure 21.4.1 provides surfaces of the scaled Riemann theta function θ ^ ( z | Ω ) , with … For the scaled Riemann theta functions depicted in Figures 21.4.221.4.5
See accompanying text
Figure 21.4.4: A real-valued scaled Riemann theta function: θ ^ ( i x , i y | Ω 1 ) , 0 x 4 , 0 y 4 . … Magnify 3D Help
See accompanying text
Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: θ ^ ( x + i y , 0 , 0 | Ω 2 ) , 0 x 1 , 0 y 3 . … Magnify 3D Help
12: 36.5 Stokes Sets
For z 0 , the Stokes set is expressed in terms of scaled coordinates …
36.5.7 X = 9 20 + 20 u 4 - Y 2 20 u 2 + 6 u 2 sign ( z ) ,
36.5.10 160 u 6 + 40 u 4 = Y 2 .
36.5.17 Y S ( X ) = Y ( u , | X | ) ,
36.5.18 f ( u , X ) = f ( - u + 1 3 , X ) ,
13: 15.14 Integrals
15.14.1 0 x s - 1 F ( a , b c ; - x ) d x = Γ ( s ) Γ ( a - s ) Γ ( b - s ) Γ ( a ) Γ ( b ) Γ ( c - s ) , min ( a , b ) > s > 0 .
14: 14.19 Toroidal (or Ring) Functions
where the constant c is a scaling factor. … With F as in §14.3 and ξ > 0 ,
14.19.2 P ν - 1 2 μ ( cosh ξ ) = Γ ( 1 2 - μ ) π 1 / 2 ( 1 - e - 2 ξ ) μ e ( ν + ( 1 / 2 ) ) ξ F ( 1 2 - μ , 1 2 + ν - μ ; 1 - 2 μ ; 1 - e - 2 ξ ) , μ 1 2 , 3 2 , 5 2 , .
15: 15.8 Transformations of Variable
15.8.1 F ( a , b c ; z ) = ( 1 - z ) - a F ( a , c - b c ; z z - 1 ) = ( 1 - z ) - b F ( c - a , b c ; z z - 1 ) = ( 1 - z ) c - a - b F ( c - a , c - b c ; z ) , | ph ( 1 - z ) | < π .
15.8.2 sin ( π ( b - a ) ) π F ( a , b c ; z ) = ( - z ) - a Γ ( b ) Γ ( c - a ) F ( a , a - c + 1 a - b + 1 ; 1 z ) - ( - z ) - b Γ ( a ) Γ ( c - b ) F ( b , b - c + 1 b - a + 1 ; 1 z ) , | ph ( - z ) | < π .
15.8.3 sin ( π ( b - a ) ) π F ( a , b c ; z ) = ( 1 - z ) - a Γ ( b ) Γ ( c - a ) F ( a , c - b a - b + 1 ; 1 1 - z ) - ( 1 - z ) - b Γ ( a ) Γ ( c - b ) F ( b , c - a b - a + 1 ; 1 1 - z ) , | ph ( - z ) | < π .
Alternatively, if b - a is a negative integer, then we interchange a and b in F ( a , b ; c ; z ) . …
15.8.12 F ( a , b ; a + b - m ; z ) = ( 1 - z ) - m F ( a ~ , b ~ ; a ~ + b ~ + m ; z ) , a ~ = a - m , b ~ = b - m .
16: 15.9 Relations to Other Functions
15.9.16 F ( a , b 2 b ; z ) = π Γ ( b ) z - b + ( 1 / 2 ) ( 1 - z ) ( b - a - ( 1 / 2 ) ) / 2 P a - b - ( 1 / 2 ) - b + ( 1 / 2 ) ( 2 - z 2 1 - z ) , b 0 , - 1 , - 2 , , | ph ( 1 - z ) | < π and | 1 - z | < 1 .
15.9.18 F ( a , b a + b + 1 2 ; z ) = 2 a + b - ( 1 / 2 ) ( - z ) ( - a - b + ( 1 / 2 ) ) / 2 P a - b - ( 1 / 2 ) - a - b + ( 1 / 2 ) ( 1 - z ) , | ph ( - z ) | < π .
15.9.20 F ( a , b 1 2 ( a + b + 1 ) ; z ) = ( - z ( 1 - z ) ) ( 1 - a - b ) / 4 P ( a - b - 1 ) / 2 ( 1 - a - b ) / 2 ( 1 - 2 z ) , | ph ( - z ) | < π .
17: 32.13 Reductions of Partial Differential Equations
has the scaling reduction … has the scaling reduction … has the scaling reduction …
18: Viewing DLMF Interactive 3D Graphics
Users can render a 3D scene and interactively rotate, scale, and otherwise explore a function surface. …
19: 15.3 Graphics
See accompanying text
Figure 15.3.7: | F ( - 3 , 3 5 ; u + i v ; 1 2 ) | , - 6 u 2 , - 2 v 2 . Magnify 3D Help
20: About Color Map
Mathematically, we scale the height to h lying in the interval [ 0 , 4 ] and the components are computed as follows … Specifically, by scaling the phase angle in [ 0 , 2 π ) to q in the interval [ 0 , 4 ) , the hue (in degrees) is computed as …