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11: 4.6 Power Series
4.6.1 ln ( 1 + z ) = z 1 2 z 2 + 1 3 z 3 , | z | 1 , z 1 ,
4.6.3 ln z = ( z 1 ) 1 2 ( z 1 ) 2 + 1 3 ( z 1 ) 3 , | z 1 | 1 , z 0 ,
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 ,
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. …
12: 22.6 Elementary Identities
22.6.2 1 + cs 2 ( z , k ) = k 2 + ds 2 ( z , k ) = ns 2 ( z , k ) ,
22.6.5 sn ( 2 z , k ) = 2 sn ( z , k ) cn ( z , k ) dn ( z , k ) 1 k 2 sn 4 ( z , k ) ,
22.6.8 cd ( 2 z , k ) = cd 2 ( z , k ) k 2 sd 2 ( z , k ) nd 2 ( z , k ) 1 + k 2 k 2 sd 4 ( z , k ) ,
22.6.9 sd ( 2 z , k ) = 2 sd ( z , k ) cd ( z , k ) nd ( z , k ) 1 + k 2 k 2 sd 4 ( z , k ) ,
22.6.10 nd ( 2 z , k ) = nd 2 ( z , k ) + k 2 sd 2 ( z , k ) cd 2 ( z , k ) 1 + k 2 k 2 sd 4 ( z , k ) ,
13: 4.21 Identities
4.21.24 sin ( z ) = sin z ,
4.21.25 cos ( z ) = cos z ,
This result is also valid when n is fractional or complex, provided that π z π . … If t = tan ( 1 2 z ) , then … With z = x + i y
14: 10.56 Generating Functions
When 2 | t | < | z | ,
10.56.1 cos z 2 2 z t z = cos z z + n = 1 t n n ! 𝗃 n 1 ( z ) ,
10.56.2 sin z 2 2 z t z = sin z z + n = 1 t n n ! 𝗒 n 1 ( z ) .
10.56.3 cosh z 2 + 2 i z t z = cosh z z + n = 1 ( i t ) n n ! 𝗂 n 1 ( 1 ) ( z ) ,
10.56.4 sinh z 2 + 2 i z t z = sinh z z + n = 1 ( i t ) n n ! 𝗂 n 1 ( 2 ) ( z ) ,
15: 32.5 Integral Equations
Let K ( z , ζ ) be the solution of
32.5.1 K ( z , ζ ) = k Ai ( z + ζ 2 ) + k 2 4 z z K ( z , s ) Ai ( s + t 2 ) Ai ( t + ζ 2 ) d s d t ,
where k is a real constant, and Ai ( z ) is defined in §9.2. …
32.5.2 w ( z ) = K ( z , z ) ,
32.5.3 w ( z ) k Ai ( z ) , z + .
16: 4.34 Derivatives and Differential Equations
4.34.1 d d z sinh z = cosh z ,
4.34.2 d d z cosh z = sinh z ,
4.34.4 d d z csch z = csch z coth z ,
4.34.5 d d z sech z = sech z tanh z ,
17: 4.1 Special Notation
The main purpose of the present chapter is to extend these definitions and properties to complex arguments z . The main functions treated in this chapter are the logarithm ln z , Ln z ; the exponential exp z , e z ; the circular trigonometric (or just trigonometric) functions sin z , cos z , tan z , csc z , sec z , cot z ; the inverse trigonometric functions arcsin z , Arcsin z , etc. ; the hyperbolic trigonometric (or just hyperbolic) functions sinh z , cosh z , tanh z , csch z , sech z , coth z ; the inverse hyperbolic functions arcsinh z , Arcsinh z , etc. Sometimes in the literature the meanings of ln and Ln are interchanged; similarly for arcsin z and Arcsin z , etc. … sin 1 z for arcsin z and Sin 1 z for Arcsin z .
18: 4.31 Special Values and Limits
Table 4.31.1: Hyperbolic functions: values at multiples of 1 2 π i .
z 0 1 2 π i π i 3 2 π i
cosh z 1 0 1 0
coth z 0 0 1
4.31.1 lim z 0 sinh z z = 1 ,
4.31.2 lim z 0 tanh z z = 1 ,
4.31.3 lim z 0 cosh z 1 z 2 = 1 2 .
19: 7.10 Derivatives
For the Hermite polynomial H n ( z ) see §18.3.
7.10.2 w ( z ) = 2 z w ( z ) + ( 2 i / π ) ,
7.10.3 w ( n + 2 ) ( z ) + 2 z w ( n + 1 ) ( z ) + 2 ( n + 1 ) w ( n ) ( z ) = 0 , n = 0 , 1 , 2 , .
d f ( z ) d z = π z g ( z ) ,
d g ( z ) d z = π z f ( z ) 1 .
20: 4.9 Continued Fractions
4.9.2 ln ( 1 + z 1 z ) = 2 z 1 z 2 3 4 z 2 5 9 z 2 7 16 z 2 9 ,
valid when z ( , 1 ] [ 1 , ) ; see Figure 4.23.1(i). … For z ,
e z = 1 1 z 1 + z 2 z 3 + z 2 z 5 + z 2
= 1 + z 1 z 2 + z 3 z 2 + z 5 z 2 + z 7