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1: 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
sinh z 0 i 0 - i
tanh z 0 i 0 - i 1
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 ,
2: 10.72 Mathematical Applications
If f ( z ) has a double zero z 0 , or more generally z 0 is a zero of order m , m = 2 , 3 , 4 , , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order 1 / ( m + 2 ) . The number m can also be replaced by any real constant λ ( > - 2 ) in the sense that ( z - z 0 ) - λ f ( z ) is analytic and nonvanishing at z 0 ; moreover, g ( z ) is permitted to have a single or double pole at z 0 . … In regions in which the function f ( z ) has a simple pole at z = z 0 and ( z - z 0 ) 2 g ( z ) is analytic at z = z 0 (the case λ = - 1 in §10.72(i)), asymptotic expansions of the solutions w of (10.72.1) for large u can be constructed in terms of Bessel functions and modified Bessel functions of order ± 1 + 4 ρ , where ρ is the limiting value of ( z - z 0 ) 2 g ( z ) as z z 0 . … In (10.72.1) assume f ( z ) = f ( z , α ) and g ( z ) = g ( z , α ) depend continuously on a real parameter α , f ( z , α ) has a simple zero z = z 0 ( α ) and a double pole z = 0 , except for a critical value α = a , where z 0 ( a ) = 0 . …These approximations are uniform with respect to both z and α , including z = z 0 ( a ) , the cut neighborhood of z = 0 , and α = a . …
3: 15.15 Sums
15.15.1 F ( a , b c ; 1 z ) = ( 1 - z 0 z ) - a s = 0 ( a ) s s ! F ( - s , b c ; 1 z 0 ) ( 1 - z z 0 ) - s .
Here z 0 ( 0 ) is an arbitrary complex constant and the expansion converges when | z - z 0 | > max ( | z 0 | , | z 0 - 1 | ) . …
4: 1.10 Functions of a Complex Variable
Then z = z 0 is an isolated singularity of f ( z ) . … For example, z = 0 is a branch point of z . … Furthermore, if g ( z ) is analytic at z 0 , then … Let w 0 = f ( z 0 ) . … Also, if in addition g ( z ) is analytic at z 0 , then …
5: 9.4 Maclaurin Series
9.4.1 Ai ( z ) = Ai ( 0 ) ( 1 + 1 3 ! z 3 + 1 4 6 ! z 6 + 1 4 7 9 ! z 9 + ) + Ai ( 0 ) ( z + 2 4 ! z 4 + 2 5 7 ! z 7 + 2 5 8 10 ! z 10 + ) ,
9.4.2 Ai ( z ) = Ai ( 0 ) ( 1 + 2 3 ! z 3 + 2 5 6 ! z 6 + 2 5 8 9 ! z 9 + ) + Ai ( 0 ) ( 1 2 ! z 2 + 1 4 5 ! z 5 + 1 4 7 8 ! z 8 + ) ,
9.4.3 Bi ( z ) = Bi ( 0 ) ( 1 + 1 3 ! z 3 + 1 4 6 ! z 6 + 1 4 7 9 ! z 9 + ) + Bi ( 0 ) ( z + 2 4 ! z 4 + 2 5 7 ! z 7 + 2 5 8 10 ! z 10 + ) ,
9.4.4 Bi ( z ) = Bi ( 0 ) ( 1 + 2 3 ! z 3 + 2 5 6 ! z 6 + 2 5 8 9 ! z 9 + ) + Bi ( 0 ) ( 1 2 ! z 2 + 1 4 5 ! z 5 + 1 4 7 8 ! z 8 + ) .
6: 20.4 Values at z = 0
§20.4 Values at z = 0
20.4.3 θ 2 ( 0 , q ) = 2 q 1 / 4 n = 1 ( 1 - q 2 n ) ( 1 + q 2 n ) 2 ,
20.4.4 θ 3 ( 0 , q ) = n = 1 ( 1 - q 2 n ) ( 1 + q 2 n - 1 ) 2 ,
20.4.5 θ 4 ( 0 , q ) = n = 1 ( 1 - q 2 n ) ( 1 - q 2 n - 1 ) 2 .
20.4.12 θ 1 ′′′ ( 0 , q ) θ 1 ( 0 , q ) = θ 2 ′′ ( 0 , q ) θ 2 ( 0 , q ) + θ 3 ′′ ( 0 , q ) θ 3 ( 0 , q ) + θ 4 ′′ ( 0 , q ) θ 4 ( 0 , q ) .
7: 4.6 Power Series
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 ,
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 .
If a = 0 , 1 , 2 , , then the series terminates and z is unrestricted. …
8: 32.4 Isomonodromy Problems
32.4.4 A ( z , λ ) = ( 4 λ 4 + 2 w 2 + z ) [ 1 0 0 - 1 ] - i ( 4 λ 2 w + 2 w 2 + z ) [ 0 - i i 0 ] - ( 2 λ w + 1 2 λ ) [ 0 1 1 0 ] ,
32.4.8 A ( z , λ ) = [ 1 4 z 0 0 - 1 4 z ] + [ - 1 2 θ u 0 u 1 1 2 θ ] 1 λ + [ v 0 - 1 4 z - v 1 v 0 ( v 0 - 1 2 z ) / v 1 1 4 z - v 0 ] 1 λ 2 ,
32.4.10 z u 0 = θ u 0 - z v 0 v 1 ,
32.4.11 z u 1 = - θ u 1 - ( z ( 2 v 0 - z ) / ( 2 v 1 ) ) ,
32.4.12 z v 0 = 2 v 0 u 1 v 1 + v 0 + ( u 0 ( 2 v 0 - z ) / v 1 ) ,
9: 29.16 Asymptotic Expansions
The approximations for Lamé polynomials hold uniformly on the rectangle 0 z K , 0 z K , when n k and n k assume large real values. …
10: 7.9 Continued Fractions
7.9.1 π e z 2 erfc z = z z 2 + 1 2 1 + 1 z 2 + 3 2 1 + 2 z 2 + , z > 0 ,
7.9.2 π e z 2 erfc z = 2 z 2 z 2 + 1 - 1 2 2 z 2 + 5 - 3 4 2 z 2 + 9 - , z > 0 ,
7.9.3 w ( z ) = i π 1 z - 1 2 z - 1 z - 3 2 z - 2 z - , z > 0 .