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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: 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 ( z ) = 𝒵 ν 1 ( z ) ( ν / z ) 𝒵 ν ( z ) ,
For results on modified quotients of the form z 𝒵 ν ± 1 ( z ) / 𝒵 ν ( z ) see Onoe (1955) and Onoe (1956). …
( 1 z d d z ) k ( z ν 𝒵 ν ( z ) ) = z ν k 𝒵 ν k ( z ) ,
3: 4.28 Definitions and Periodicity
4.28.9 cos ( i z ) = cosh z ,
4.28.11 csc ( i z ) = i csch 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 .
4: 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 …
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: 22.13 Derivatives and Differential Equations
22.13.1 ( d d z sn ( z , k ) ) 2 = ( 1 sn 2 ( z , k ) ) ( 1 k 2 sn 2 ( z , k ) ) ,
22.13.4 ( d d z cd ( z , k ) ) 2 = ( 1 cd 2 ( z , k ) ) ( 1 k 2 cd 2 ( z , k ) ) ,
22.13.7 ( d d z dc ( z , k ) ) 2 = ( dc 2 ( z , k ) 1 ) ( dc 2 ( z , k ) k 2 ) ,
22.13.10 ( d d z ns ( z , k ) ) 2 = ( ns 2 ( z , k ) k 2 ) ( ns 2 ( z , k ) 1 ) ,
22.13.19 d 2 d z 2 dc ( z , k ) = ( 1 + k 2 ) dc ( z , k ) + 2 dc 3 ( z , k ) ,
7: 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 ) ,
8: 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 ) ,
9: 1.13 Differential Equations
The equation … Two solutions w 1 ( z ) and w 2 ( z ) are called a fundamental pair if any other solution w ( z ) is expressible as … where c is independent of z and f ( z ) is defined in (1.13.1). … The inhomogeneous (or nonhomogeneous) equation … Let W ( z ) satisfy (1.13.14), ζ ( z ) be any thrice-differentiable function of z , and …
10: 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 ,