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11: 4.28 Definitions and Periodicity
4.28.2 cosh z = e z + e - z 2 ,
4.28.3 cosh z ± sinh z = e ± z ,
4.28.5 csch z = 1 sinh z ,
4.28.9 cos ( i z ) = cosh z ,
4.28.12 sec ( i z ) = sech z ,
12: 4.7 Derivatives and Differential Equations
4.7.6 w ( z ) = Ln ( f ( z ) ) +  constant .
4.7.10 d d z z a = a z a - 1 ,
4.7.12 d w d z = f ( z ) w
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 ,
13: 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 ) .
14: 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 ) ,
15: 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 ) .
16: 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 .
17: 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 .
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: 4.33 Maclaurin Series and Laurent Series
4.33.1 sinh z = z + z 3 3 ! + z 5 5 ! + ,
4.33.2 cosh z = 1 + z 2 2 ! + z 4 4 ! + .
4.33.3 tanh z = z - z 3 3 + 2 15 z 5 - 17 315 z 7 + + 2 2 n ( 2 2 n - 1 ) B 2 n ( 2 n ) ! z 2 n - 1 + , | z | < 1 2 π .