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1: 31.2 Differential Equations
Jacobi’s Elliptic Form
2: 22.5 Special Values
For example, at z = K + i K , sn ( z , k ) = 1 / k , d sn ( z , k ) / d z = 0 . … Table 22.5.2 gives sn ( z , k ) , cn ( z , k ) , dn ( z , k ) for other special values of z . For example, sn ( 1 2 K , k ) = ( 1 + k ) - 1 / 2 . …
§22.5(ii) Limiting Values of k
3: Errata
  • Table 22.5.4


    Originally the limiting form for sc ( z , k ) in the last line of this table was incorrect ( cosh z , instead of sinh z ).

    sn ( z , k ) tanh z cd ( z , k ) 1 dc ( z , k ) 1 ns ( z , k ) coth z
    cn ( z , k ) sech z sd ( z , k ) sinh z nc ( z , k ) cosh z ds ( z , k ) csch z
    dn ( z , k ) sech z nd ( z , k ) cosh z sc ( z , k ) sinh z cs ( z , k ) csch z

    Reported 2010-11-23.

  • 4: 22.18 Mathematical Applications
    For any two points ( x 1 , y 1 ) and ( x 2 , y 2 ) on this curve, their sum ( x 3 , y 3 ) , always a third point on the curve, is defined by the Jacobi–Abel addition law …
    5: 29.12 Definitions
    With the substitution ξ = sn 2 ( z , k ) every Lamé polynomial in Table 29.12.1 can be written in the form
    6: 29.15 Fourier Series and Chebyshev Series
    Since (29.2.5) implies that cos ϕ = sn ( z , k ) , (29.15.1) can be rewritten in the form
    7: 22.8 Addition Theorems
    §22.8 Addition Theorems
    §22.8(ii) Alternative Forms for Sum of Two Arguments
    §22.8(iii) Special Relations Between Arguments
    If sums/differences of the z j ’s are rational multiples of K ( k ) , then further relations follow. …
    8: 29.2 Differential Equations
    For sn ( z , k ) see §22.2. …
    §29.2(ii) Other Forms
    For am ( z , k ) see §22.16(i). … we have …
    9: 22.15 Inverse Functions
    §22.15 Inverse Functions
    are denoted respectively by …
    §22.15(ii) Representations as Elliptic Integrals
    The integrals (22.15.12)–(22.15.14) can be regarded as normal forms for representing the inverse functions. …
    10: 22.4 Periods, Poles, and Zeros
    For example, the poles of sn ( z , k ) , abbreviated as sn in the following tables, are at z = 2 m K + ( 2 n + 1 ) i K . … Again, one member of each congruent set of zeros appears in the second row; all others are generated by translations of the form 2 m K + 2 n i K , where m , n . … Then: (a) In any lattice unit cell p q ( z , k ) has a simple zero at z = p and a simple pole at z = q . (b) The difference between p and the nearest q is a half-period of p q ( z , k ) . … For example, sn ( z + K , k ) = cd ( z , k ) . …