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1: 4.14 Definitions and Periodicity
4.14.4 tan z = sin z cos z ,
4.14.5 csc z = 1 sin z ,
The functions sin z and cos z are entire. In the zeros of sin z are z = k π , k ; the zeros of cos z are z = ( k + 1 2 ) π , k . The functions tan z , csc z , sec z , and cot z are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7). …
2: 4.28 Definitions and Periodicity
4.28.2 cosh z = e z + e - z 2 ,
4.28.9 cos ( i z ) = cosh z ,
4.28.12 sec ( i z ) = sech z ,
The functions sinh z and cosh z have period 2 π i , and tanh z has period π i . The zeros of sinh z and cosh z are z = i k π and z = i ( k + 1 2 ) π , respectively, k .
3: 10.51 Recurrence Relations and Derivatives
Let f n ( z ) denote any of j n ( z ) , y n ( z ) , h n ( 1 ) ( z ) , or h n ( 2 ) ( z ) . …
( 1 z d d z ) m ( z n + 1 f n ( z ) ) = z n - m + 1 f n - m ( z ) , m = 0 , 1 , , n ,
( 1 z d d z ) m ( z - n f n ( z ) ) = ( - 1 ) m z - n - m f n + m ( z ) , m = 0 , 1 , .
Let g n ( z ) denote i n ( 1 ) ( z ) , i n ( 2 ) ( z ) , or ( - 1 ) n k n ( z ) . Then …
4: 10.29 Recurrence Relations and Derivatives
With 𝒵 ν ( z ) defined as in §10.25(ii),
𝒵 ν - 1 ( z ) - 𝒵 ν + 1 ( z ) = ( 2 ν / z ) 𝒵 ν ( z ) ,
𝒵 ν - 1 ( z ) + 𝒵 ν + 1 ( z ) = 2 𝒵 ν ( z ) .
𝒵 ν ( z ) = 𝒵 ν - 1 ( z ) - ( ν / z ) 𝒵 ν ( z ) ,
𝒵 ν ( z ) = 𝒵 ν + 1 ( z ) + ( ν / z ) 𝒵 ν ( z ) .
5: 10.36 Other Differential Equations
The quantity λ 2 in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by - λ 2 if at the same time the symbol 𝒞 in the given solutions is replaced by 𝒵 . …
10.36.1 z 2 ( z 2 + ν 2 ) w ′′ + z ( z 2 + 3 ν 2 ) w - ( ( z 2 + ν 2 ) 2 + z 2 - ν 2 ) w = 0 , w = 𝒵 ν ( z ) ,
10.36.2 z 2 w ′′ + z ( 1 ± 2 z ) w + ( ± z - ν 2 ) w = 0 , w = e z 𝒵 ν ( z ) .
Differential equations for products can be obtained from (10.13.9)–(10.13.11) by replacing z by i z .
6: 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 ) .
C ( - z ) = - C ( z ) ,
7: 22.10 Maclaurin Series
§22.10(i) Maclaurin Series in z
The full expansions converge when | z | < min ( K ( k ) , K ( k ) ) . …
22.10.4 sn ( z , k ) = sin z - k 2 4 ( z - sin z cos z ) cos z + O ( k 4 ) ,
22.10.5 cn ( z , k ) = cos z + k 2 4 ( z - sin z cos z ) sin z + O ( k 4 ) ,
22.10.8 cn ( z , k ) = sech z + k 2 4 ( z - sinh z cosh z ) tanh z sech z + O ( k 4 ) ,
8: 4.20 Derivatives and Differential Equations
4.20.1 d d z sin z = cos z ,
4.20.2 d d z cos z = - sin z ,
4.20.3 d d z tan z = sec 2 z ,
4.20.4 d d z csc z = - csc z cot z ,
4.20.5 d d z sec z = sec z tan z ,
9: 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 .
For expansions that correspond to (4.19.4)–(4.19.9), change z to i z and use (4.28.8)–(4.28.13).
10: 4.6 Power Series
4.6.2 ln z = ( z - 1 z ) + 1 2 ( z - 1 z ) 2 + 1 3 ( z - 1 z ) 3 + , z 1 2 ,
4.6.4 ln z = 2 ( ( z - 1 z + 1 ) + 1 3 ( z - 1 z + 1 ) 3 + 1 5 ( z - 1 z + 1 ) 5 + ) , z 0 , z 0 ,
4.6.6 ln ( z + a ) = ln a + 2 ( ( z 2 a + z ) + 1 3 ( z 2 a + z ) 3 + 1 5 ( z 2 a + z ) 5 + ) , a > 0 , z - a , z - a .
valid when a is any real or complex constant and | z | < 1 . If a = 0 , 1 , 2 , , then the series terminates and z is unrestricted. …