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21: 4.8 Identities
4.8.2 ln ( z 1 z 2 ) = ln z 1 + ln z 2 , - π ph z 1 + ph z 2 π ,
4.8.7 ln 1 z = - ln z , | ph z | π .
4.8.15 a z b z = ( a b ) z , - π ph a + ph b π ,
22: 7.2 Definitions
7.2.1 erf z = 2 π 0 z e - t 2 d t ,
7.2.2 erfc z = 2 π z e - t 2 d t = 1 - erf z ,
7.2.6 ( z ) = z e 1 2 π i t 2 d t ,
7.2.7 C ( z ) = 0 z cos ( 1 2 π t 2 ) d t ,
7.2.8 S ( z ) = 0 z sin ( 1 2 π t 2 ) d t ,
23: 7.6 Series Expansions
7.6.1 erf z = 2 π n = 0 ( - 1 ) n z 2 n + 1 n ! ( 2 n + 1 ) ,
7.6.2 erf z = 2 π e - z 2 n = 0 2 n z 2 n + 1 1 3 ( 2 n + 1 ) ,
7.6.4 C ( z ) = n = 0 ( - 1 ) n ( 1 2 π ) 2 n ( 2 n ) ! ( 4 n + 1 ) z 4 n + 1 ,
7.6.6 S ( z ) = n = 0 ( - 1 ) n ( 1 2 π ) 2 n + 1 ( 2 n + 1 ) ! ( 4 n + 3 ) z 4 n + 3 ,
7.6.10 C ( z ) = z n = 0 j 2 n ( 1 2 π z 2 ) ,
24: 25.4 Reflection Formulas
25.4.1 ζ ( 1 - s ) = 2 ( 2 π ) - s cos ( 1 2 π s ) Γ ( s ) ζ ( s ) ,
25.4.2 ζ ( s ) = 2 ( 2 π ) s - 1 sin ( 1 2 π s ) Γ ( 1 - s ) ζ ( 1 - s ) .
25.4.4 ξ ( s ) = 1 2 s ( s - 1 ) Γ ( 1 2 s ) π - s / 2 ζ ( s ) .
25.4.5 ( - 1 ) k ζ ( k ) ( 1 - s ) = 2 ( 2 π ) s m = 0 k r = 0 m ( k m ) ( m r ) ( ( c k - m ) cos ( 1 2 π s ) + ( c k - m ) sin ( 1 2 π s ) ) Γ ( r ) ( s ) ζ ( m - r ) ( s ) ,
25.4.6 c - ln ( 2 π ) - 1 2 π i .
25: 7.5 Interrelations
7.5.1 F ( z ) = 1 2 i π ( e - z 2 - w ( z ) ) = - 1 2 i π e - z 2 erf ( i z ) .
7.5.3 C ( z ) = 1 2 + f ( z ) sin ( 1 2 π z 2 ) - g ( z ) cos ( 1 2 π z 2 ) ,
7.5.7 ζ = 1 2 π ( 1 i ) z ,
7.5.13 G ( x ) = π F ( x ) - 1 2 e - x 2 Ei ( x 2 ) , x > 0 .
26: 11.4 Basic Properties
11.4.5 H 1 2 ( z ) = ( 2 π z ) 1 2 ( 1 - cos z ) ,
11.4.6 H - 1 2 ( z ) = ( 2 π z ) 1 2 sin z ,
11.4.7 L 1 2 ( z ) = ( 2 π z ) 1 2 ( cosh z - 1 ) ,
11.4.10 H - 3 2 ( z ) = ( 2 π z ) 1 2 ( cos z - sin z z ) ,
27: 22.11 Fourier and Hyperbolic Series
22.11.1 sn ( z , k ) = 2 π K k n = 0 q n + 1 2 sin ( ( 2 n + 1 ) ζ ) 1 - q 2 n + 1 ,
22.11.2 cn ( z , k ) = 2 π K k n = 0 q n + 1 2 cos ( ( 2 n + 1 ) ζ ) 1 + q 2 n + 1 ,
22.11.3 dn ( z , k ) = π 2 K + 2 π K n = 1 q n cos ( 2 n ζ ) 1 + q 2 n .
22.11.4 cd ( z , k ) = 2 π K k n = 0 ( - 1 ) n q n + 1 2 cos ( ( 2 n + 1 ) ζ ) 1 - q 2 n + 1 ,
22.11.5 sd ( z , k ) = 2 π K k k n = 0 ( - 1 ) n q n + 1 2 sin ( ( 2 n + 1 ) ζ ) 1 + q 2 n + 1 ,
28: 10.34 Analytic Continuation
10.34.1 I ν ( z e m π i ) = e m ν π i I ν ( z ) ,
10.34.2 K ν ( z e m π i ) = e - m ν π i K ν ( z ) - π i sin ( m ν π ) csc ( ν π ) I ν ( z ) .
10.34.4 K ν ( z e m π i ) = csc ( ν π ) ( ± sin ( m ν π ) K ν ( z e ± π i ) sin ( ( m 1 ) ν π ) K ν ( z ) ) .
10.34.5 K n ( z e m π i ) = ( - 1 ) m n K n ( z ) + ( - 1 ) n ( m - 1 ) - 1 m π i I n ( z ) ,
10.34.6 K n ( z e m π i ) = ± ( - 1 ) n ( m - 1 ) m K n ( z e ± π i ) ( - 1 ) n m ( m 1 ) K n ( z ) .
29: 11.7 Integrals and Sums
11.7.2 z - ν H ν + 1 ( z ) d z = - z - ν H ν ( z ) + 2 - ν z π Γ ( ν + 3 2 ) ,
11.7.4 z - ν L ν + 1 ( z ) d z = z - ν L ν ( z ) - 2 - ν z π Γ ( ν + 3 2 ) .
11.7.6 f ν + 1 ( z ) = ( 2 ν + 1 ) f ν ( z ) - z ν + 1 H ν ( z ) + ( 1 2 z 2 ) ν + 1 ( ν + 1 ) π Γ ( ν + 3 2 ) , ν > - 1 .
11.7.9 0 H ν ( t ) d t = - cot ( 1 2 π ν ) , - 2 < ν < 0 ,
11.7.10 0 t - ν - 1 H ν ( t ) d t = π 2 ν + 1 Γ ( ν + 1 ) , ν > - 3 2 ,
30: 20.3 Graphics