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11: 22.4 Periods, Poles, and Zeros
Table 22.4.1: Periods and poles of Jacobian elliptic functions.
Periods z -Poles
i K K + i K K 0
Table 22.4.2: Periods and zeros of Jacobian elliptic functions.
Periods z -Zeros
0 K K + i K i K
Figure 22.4.1 illustrates the locations in the z -plane of the poles and zeros of the three principal Jacobian functions in the rectangle with vertices 0 , 2 K , 2 K + 2 i K , 2 i K . … Figure 22.4.2 depicts the fundamental unit cell in the z -plane, with vertices s = 0 , c = K , d = K + i K , n = i K . … This half-period will be plus or minus a member of the triple K , i K , K + i K ; the other two members of this triple are quarter periods of p q ( z , k ) . …
12: 19.13 Integrals of Elliptic Integrals
§19.13 Integrals of Elliptic Integrals
§19.13(i) Integration with Respect to the Modulus
Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for F ( ϕ , k ) and E ( ϕ , k ) , together with special cases.
§19.13(iii) Laplace Transforms
For direct and inverse Laplace transforms for the complete elliptic integrals K ( k ) , E ( k ) , and D ( k ) see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.
13: 29.17 Other Solutions
29.17.1 F ( z ) = E ( z ) i K z d u ( E ( u ) ) 2 .
They are algebraic functions of sn ( z , k ) , cn ( z , k ) , and dn ( z , k ) , and have primitive period 8 K . … Lamé–Wangerin functions are solutions of (29.2.1) with the property that ( sn ( z , k ) ) 1 / 2 w ( z ) is bounded on the line segment from i K to 2 K + i K . …
14: 22.11 Fourier and Hyperbolic Series
22.11.3 dn ( z , k ) = π 2 K + 2 π K n = 1 q n cos ( 2 n ζ ) 1 + q 2 n .
Next, with E = E ( k ) denoting the complete elliptic integral of the second kind (§19.2(ii)) and q exp ( 2 | ζ | ) < 1 , where E = E ( k ) is defined by §19.2.9. …
15: 19.3 Graphics
§19.3 Graphics
See accompanying text
Figure 19.3.1: K ( k ) and E ( k ) as functions of k 2 for - 2 k 2 1 . Graphs of K ( k ) and E ( k ) are the mirror images in the vertical line k 2 = 1 2 . Magnify
See accompanying text
Figure 19.3.5: Π ( α 2 , k ) as a function of k 2 and α 2 for - 2 k 2 < 1 , - 2 α 2 2 . …If α 2 = 0 , then it reduces to K ( k ) . … Magnify 3D Help
See accompanying text
Figure 19.3.12: ( E ( k ) ) as a function of complex k 2 for - 2 ( k 2 ) 2 , - 2 ( k 2 ) 2 . … Magnify 3D Help
16: 19.38 Approximations
§19.38 Approximations
Minimax polynomial approximations (§3.11(i)) for K ( k ) and E ( k ) in terms of m = k 2 with 0 m < 1 can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for K ( k ) and E ( k ) for 0 < k 1 are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. …
17: 19.4 Derivatives and Differential Equations
§19.4(i) Derivatives
d ( E ( k ) - k 2 K ( k ) ) d k = k K ( k ) ,
d E ( k ) d k = E ( k ) - K ( k ) k ,
d ( E ( k ) - K ( k ) ) d k = - k E ( k ) k 2 ,
If ϕ = π / 2 , then these two equations become hypergeometric differential equations (15.10.1) for K ( k ) and E ( k ) . …
18: 29.14 Orthogonality
29.14.2 g , h = 0 K 0 K w ( s , t ) g ( s , t ) h ( s , t ) d t d s ,
29.14.3 w ( s , t ) = sn 2 ( K + i t , k ) - sn 2 ( s , k ) .
29.14.4 sE 2 n + 1 m ( s , k 2 ) sE 2 n + 1 m ( K + i t , k 2 ) ,
29.14.5 cE 2 n + 1 m ( s , k 2 ) cE 2 n + 1 m ( K + i t , k 2 ) ,
29.14.11 g , h = 0 4 K 0 2 K w ( s , t ) g ( s , t ) h ( s , t ) d t d s ,
19: 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. …
20: 19.1 Special Notation
l , m , n

nonnegative integers.

K ( k ) ,
E ( k ) ,
F ( ϕ , k ) ,
E ( ϕ , k ) ,