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11: 30.15 Signal Analysis
§30.15(i) Scaled Spheroidal Wave Functions
§30.15(ii) Integral Equation
§30.15(iv) Orthogonality
§30.15(v) Extremal Properties
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: 5.11 Asymptotic Expansions
5.11.3 Γ ( z ) = e z z z ( 2 π z ) 1 / 2 Γ ( z ) e z z z ( 2 π z ) 1 / 2 k = 0 g k z k ,
The scaled gamma function Γ ( z ) is defined in (5.11.3) and its main property is Γ ( z ) 1 as z in the sector | ph z | π δ . …
14: 8.18 Asymptotic Expansions of I x ( a , b )
General Case
For the scaled gamma function Γ ( z ) see (5.11.3). …
15: 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
16: 5.9 Integral Representations
5.9.11_1 Γ ( z ) = 1 1 2 π i 0 e 2 π t Γ ( t e i π / 2 ) t + i z d t + 1 2 π i 0 e 2 π t Γ ( t e i π / 2 ) t i z d t ,
5.9.11_2 1 Γ ( z ) = 1 1 2 π i 0 e 2 π t Γ ( t e i π / 2 ) t i z d t + 1 2 π i 0 e 2 π t Γ ( t e i π / 2 ) t + i z d t ,
where | ph z | < π / 2 , and the scaled gamma function Γ ( z ) is defined in (5.11.3). …
17: 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 , .
18: 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 .
19: 15.9 Relations to Other Functions
15.9.17 𝐅 ( a , a + 1 2 c ; z ) = 2 c 1 z ( 1 c ) / 2 ( 1 z ) a + ( ( c 1 ) / 2 ) P 2 a c 1 c ( 1 1 z ) , | ph z | < π and | ph ( 1 z ) | < π .
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 .
20: 32.13 Reductions of Partial Differential Equations
has the scaling reduction … has the scaling reduction … has the scaling reduction …