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11: 6.9 Continued Fraction
6.9.1 E 1 ( z ) = e - z z + 1 1 + 1 z + 2 1 + 2 z + 3 1 + 3 z + , | ph z | < π .
12: 15.19 Methods of Computation
This is because the linear transformations map the pair { e π i / 3 , e - π i / 3 } onto itself. …
13: 10.37 Inequalities; Monotonicity
If 0 ν < μ and | ph z | 1 2 π , then …Note that previously we did mention that (10.37.1) holds for | ph z | < π . This is definitely not the case. …
14: 1.10 Functions of a Complex Variable
If D = ( - , 0 ] and z = r e i θ , then one branch is r e i θ / 2 , the other branch is - r e i θ / 2 , with - π < θ < π in both cases. Similarly if D = [ 0 , ) , then one branch is r e i θ / 2 , the other branch is - r e i θ / 2 , with 0 < θ < 2 π in both cases. …
15: 19.6 Special Cases
Circular and hyperbolic cases, including Cauchy principal values, are unified by using R C ( x , y ) . …
16: 16.5 Integral Representations and Integrals
In the case p = q the left-hand side of (16.5.1) is an entire function, and the right-hand side supplies an integral representation valid when | ph ( - z ) | < π / 2 . In the case p = q + 1 the right-hand side of (16.5.1) supplies the analytic continuation of the left-hand side from the open unit disk to the sector | ph ( 1 - z ) | < π ; compare §16.2(iii). …
17: 22.5 Special Values
For values of K , K when k 2 = 1 2 (lemniscatic case) see §23.5(iii), and for k 2 = e i π / 3 (equianharmonic case) see §23.5(v). …
18: 15.9 Relations to Other Functions
For the case 0 < z < 1 see (14.3.1). …
15.9.26 D ( k ) = π 4 F ( 1 2 , 3 2 2 ; k 2 ) .
19: 10.41 Asymptotic Expansions for Large Order
We then extend the validity of this property from z ± i to z in the sector - π + δ ph z 2 π - δ in the case of H ν ( 1 ) ( ν z ) , and to z in the sector - 2 π + δ ph z π - δ in the case of H ν ( 2 ) ( ν z ) . …
20: 28.29 Definitions and Basic Properties
The case c = 0 is equivalent to …The solutions of period π or 2 π are exceptional in the following sense. … In the symmetric case Q ( z ) = Q ( - z ) , w I ( z , λ ) is an even solution and w II ( z , λ ) is an odd solution; compare §28.2(ii). …The cases ν = 0 and ν = 1 split into four subcases as in (28.2.21) and (28.2.22). …
§28.29(iii) Discriminant and Eigenvalues in the Real Case