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11: 4.7 Derivatives and Differential Equations
4.7.1 d d z ln z = 1 z ,
4.7.6 w ( z ) = Ln ( f ( z ) ) +  constant .
4.7.10 d d z z a = a z a 1 ,
4.7.14 d 2 w d z 2 = a w , a 0 ,
4.7.15 w = A e a z + B e a z ,
12: 5.18 q -Gamma and q -Beta Functions
5.18.1 ( a ; q ) n = k = 0 n 1 ( 1 a q k ) , n = 0 , 1 , 2 , ,
5.18.3 ( a ; q ) = k = 0 ( 1 a q k ) .
5.18.7 Γ q ( z + 1 ) = 1 q z 1 q Γ q ( z ) .
5.18.10 lim q 1 Γ q ( z ) = Γ ( z ) .
5.18.11 B q ( a , b ) = Γ q ( a ) Γ q ( b ) Γ q ( a + b ) .
13: 28.19 Expansions in Series of me ν + 2 n Functions
28.19.2 f ( z ) = n = f n me ν + 2 n ( z , q ) ,
28.19.3 f n = 1 π 0 π f ( z ) me ν + 2 n ( z , q ) d z .
28.19.4 e i ν z = n = c 2 n ν + 2 n ( q ) me ν + 2 n ( z , q ) ,
14: 7.4 Symmetry
7.4.1 erf ( z ) = erf ( z ) ,
7.4.2 erfc ( z ) = 2 erfc ( z ) ,
7.4.3 w ( z ) = 2 e z 2 w ( z ) .
7.4.4 F ( z ) = F ( z ) .
15: 4.22 Infinite Products and Partial Fractions
4.22.1 sin z = z n = 1 ( 1 z 2 n 2 π 2 ) ,
4.22.2 cos z = n = 1 ( 1 4 z 2 ( 2 n 1 ) 2 π 2 ) .
4.22.3 cot z = 1 z + 2 z n = 1 1 z 2 n 2 π 2 ,
4.22.4 csc 2 z = n = 1 ( z n π ) 2 ,
4.22.5 csc z = 1 z + 2 z n = 1 ( 1 ) n z 2 n 2 π 2 .
16: 4.36 Infinite Products and Partial Fractions
4.36.1 sinh z = z n = 1 ( 1 + z 2 n 2 π 2 ) ,
4.36.2 cosh z = n = 1 ( 1 + 4 z 2 ( 2 n 1 ) 2 π 2 ) .
4.36.3 coth z = 1 z + 2 z n = 1 1 z 2 + n 2 π 2 ,
4.36.4 csch 2 z = n = 1 ( z n π i ) 2 ,
4.36.5 csch z = 1 z + 2 z n = 1 ( 1 ) n z 2 + n 2 π 2 .
17: 10.13 Other Differential Equations
10.13.1 w ′′ + ( λ 2 ν 2 1 4 z 2 ) w = 0 , w = z 1 2 𝒞 ν ( λ z ) ,
10.13.2 w ′′ + ( λ 2 4 z ν 2 1 4 z 2 ) w = 0 , w = z 1 2 𝒞 ν ( λ z 1 2 ) ,
10.13.5 z 2 w ′′ + ( 1 2 r ) z w + ( λ 2 q 2 z 2 q + r 2 ν 2 q 2 ) w = 0 , w = z r 𝒞 ν ( λ z q ) ,
10.13.9 z 2 w ′′′ + 3 z w ′′ + ( 4 z 2 + 1 4 ν 2 ) w + 4 z w = 0 , w = 𝒞 ν ( z ) 𝒟 ν ( z ) ,
10.13.10 z 3 w ′′′ + z ( 4 z 2 + 1 4 ν 2 ) w + ( 4 ν 2 1 ) w = 0 , w = z 𝒞 ν ( z ) 𝒟 ν ( z ) ,
18: 12.8 Recurrence Relations and Derivatives
12.8.1 z U ( a , z ) U ( a 1 , z ) + ( a + 1 2 ) U ( a + 1 , z ) = 0 ,
12.8.2 U ( a , z ) + 1 2 z U ( a , z ) + ( a + 1 2 ) U ( a + 1 , z ) = 0 ,
12.8.3 U ( a , z ) 1 2 z U ( a , z ) + U ( a 1 , z ) = 0 ,
12.8.4 2 U ( a , z ) + U ( a 1 , z ) + ( a + 1 2 ) U ( a + 1 , z ) = 0 .
12.8.5 z V ( a , z ) V ( a + 1 , z ) + ( a 1 2 ) V ( a 1 , z ) = 0 ,
19: 5.12 Beta Function
5.12.1 B ( a , b ) = 0 1 t a 1 ( 1 t ) b 1 d t = Γ ( a ) Γ ( b ) Γ ( a + b ) .
5.12.3 0 t a 1 d t ( 1 + t ) a + b = B ( a , b ) .
5.12.4 0 1 t a 1 ( 1 t ) b 1 ( t + z ) a + b d t = B ( a , b ) ( 1 + z ) a z b , | ph z | < π .
5.12.8 1 2 π d t ( w + i t ) a ( z i t ) b = ( w + z ) 1 a b ( a + b 1 ) B ( a , b ) , ( a + b ) > 1 , w > 0 , z > 0 .
5.12.12 P ( 1 + , 0 + , 1 , 0 ) t a 1 ( 1 t ) b 1 d t = 4 e π i ( a + b ) sin ( π a ) sin ( π b ) B ( a , b ) ,
20: 4.21 Identities
4.21.12 sin 2 z + cos 2 z = 1 ,
4.21.13 sec 2 z = 1 + tan 2 z ,
4.21.14 csc 2 z = 1 + cot 2 z .
4.21.24 sin ( z ) = sin z ,
4.21.25 cos ( z ) = cos z ,