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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: 1.1 Special Notation
x , y real variables.
z complex variable in §§1.2(i), 1.91.11, real variable in §§1.51.6.
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 ,
27.4.7 n = 1 λ ( n ) n - s = ζ ( 2 s ) ζ ( s ) , 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 ,
9: 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
10: 5.1 Special Notation
j , m , n nonnegative integers.
x , y real variables.
a , b , q , s , w real or complex variables with | q | < 1 .