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1: 32.1 Special Notation
m , n integers.
x real variable.
z complex variable.
k real parameter.
2: 5.24 Software
§5.24(ii) Γ ( x ) , x
§5.24(iii) ψ ( x ) , ψ ( n ) ( x ) , x
§5.24(v) B ( a , b ) , a , b
3: 4.3 Graphics
§4.3(i) Real Arguments
See accompanying text
Figure 4.3.1: ln x and e x . … Magnify
Figure 4.3.2 illustrates the conformal mapping of the strip π < z < π onto the whole w -plane cut along the negative real axis, where w = e z and z = ln w (principal value). …Lines parallel to the real axis in the z -plane map onto rays in the w -plane, and lines parallel to the imaginary axis in the z -plane map onto circles centered at the origin in the w -plane. In the labeling of corresponding points r is a real parameter that can lie anywhere in the interval ( 0 , ) . …
4: 15.7 Continued Fractions
15.7.1 𝐅 ( a , b ; c ; z ) 𝐅 ( a , b + 1 ; c + 1 ; z ) = t 0 u 1 z t 1 u 2 z t 2 u 3 z t 3 ,
If z < 1 2 , then
5: 8.14 Integrals
8.14.1 0 e a x γ ( b , x ) Γ ( b ) d x = ( 1 + a ) b a , a > 0 , b > 1 ,
8.14.2 0 e a x Γ ( b , x ) d x = Γ ( b ) 1 ( 1 + a ) b a , a > 1 , b > 1 .
8.14.3 0 x a 1 γ ( b , x ) d x = Γ ( a + b ) a , a < 0 , ( a + b ) > 0 ,
8.14.4 0 x a 1 Γ ( b , x ) d x = Γ ( a + b ) a , a > 0 , ( a + b ) > 0 ,
8.14.6 0 x a 1 e s x Γ ( b , x ) d x = Γ ( a + b ) a ( 1 + s ) a + b F ( 1 , a + b ; 1 + a ; s / ( 1 + s ) ) , s > 1 , ( a + b ) > 0 , a > 0 .
6: 27.4 Euler Products and Dirichlet Series
27.4.3 ζ ( s ) = n = 1 n s = p ( 1 p s ) 1 , s > 1 .
27.4.5 n = 1 μ ( n ) n s = 1 ζ ( s ) , s > 1 ,
27.4.6 n = 1 ϕ ( n ) n s = ζ ( s 1 ) ζ ( s ) , s > 2 ,
27.4.10 n = 1 d k ( n ) n s = ( ζ ( s ) ) k , s > 1 ,
27.4.11 n = 1 σ α ( n ) n s = ζ ( s ) ζ ( s α ) , s > max ( 1 , 1 + α ) ,
7: 4.29 Graphics
§4.29(i) Real Arguments
See accompanying text
Figure 4.29.6: Principal values of arccsch x and arcsech x . … Magnify
8: 31.3 Basic Solutions
31.3.2 a γ c 1 q c 0 = 0 ,
31.3.7 ( 1 z ) 1 δ H ( 1 a , ( ( 1 a ) γ + ϵ ) ( 1 δ ) + α β q ; α + 1 δ , β + 1 δ , 2 δ , γ ; 1 z ) .
9: 5.13 Integrals
5.13.1 1 2 π i c i c + i Γ ( s + a ) Γ ( b s ) z s d s = Γ ( a + b ) z a ( 1 + z ) a + b , ( a + b ) > 0 , a < c < b , | ph z | < π .
5.13.2 1 2 π | Γ ( a + i t ) | 2 e ( 2 b π ) t d t = Γ ( 2 a ) ( 2 sin b ) 2 a , a > 0 , 0 < b < π .
5.13.3 1 2 π Γ ( a + i t ) Γ ( b + i t ) Γ ( c i t ) Γ ( d i t ) d t = Γ ( a + c ) Γ ( a + d ) Γ ( b + c ) Γ ( b + d ) Γ ( a + b + c + d ) , a , b , c , d > 0 .
5.13.4 d t Γ ( a + t ) Γ ( b + t ) Γ ( c t ) Γ ( d t ) = Γ ( a + b + c + d 3 ) Γ ( a + c 1 ) Γ ( a + d 1 ) Γ ( b + c 1 ) Γ ( b + d 1 ) , ( a + b + c + d ) > 3 .
5.13.5 1 4 π k = 1 4 Γ ( a k + i t ) Γ ( a k i t ) Γ ( 2 i t ) Γ ( 2 i t ) d t = 1 j < k 4 Γ ( a j + a k ) Γ ( a 1 + a 2 + a 3 + a 4 ) , ( a k ) > 0 , k = 1 , 2 , 3 , 4 .
10: 4.15 Graphics
§4.15(i) Real Arguments
Figure 4.15.7 illustrates the conformal mapping of the strip 1 2 π < z < 1 2 π onto the whole w -plane cut along the real axis from to 1 and 1 to , where w = sin z and z = arcsin w (principal value). …Lines parallel to the real axis in the z -plane map onto ellipses in the w -plane with foci at w = ± 1 , and lines parallel to the imaginary axis in the z -plane map onto rectangular hyperbolas confocal with the ellipses. In the labeling of corresponding points r is a real parameter that can lie anywhere in the interval ( 0 , ) . …
See accompanying text
Figure 4.15.13: arccsc ( x + i y ) (principal value). There is a branch cut along the real axis from 1 to 1 . Magnify 3D Help