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1: 22.16 Related Functions
§22.16(i) Jacobi’s Amplitude ( am ) Function
Definition
Quasi-Periodicity
Relation to Elliptic Integrals
See accompanying text
Figure 22.16.1: Jacobi’s amplitude function am ( x , k ) for 0 x 10 π and k = 0.4 , 0.7 , 0.99 , 0.999999 . … Magnify
2: 22.21 Tables
§22.21 Tables
3: 22.19 Physical Applications
§22.19(i) Classical Dynamics: The Pendulum
22.19.3 θ ( t ) = 2 am ( t E / 2 , 2 / E ) ,
See accompanying text
Figure 22.19.1: Jacobi’s amplitude function am ( x , k ) for 0 x 10 π and k = 0.5 , 0.9999 , 1.0001 , 2 . … Magnify
4: 22.1 Special Notation
The functions treated in this chapter are the three principal Jacobian elliptic functions sn ( z , k ) , cn ( z , k ) , dn ( z , k ) ; the nine subsidiary Jacobian elliptic functions cd ( z , k ) , sd ( z , k ) , nd ( z , k ) , dc ( z , k ) , nc ( z , k ) , sc ( z , k ) , ns ( z , k ) , ds ( z , k ) , cs ( z , k ) ; the amplitude function am ( x , k ) ; Jacobi’s epsilon and zeta functions ( x , k ) and Z ( x | k ) . …
5: 22.20 Methods of Computation
§22.20(vi) Related Functions
6: 19.25 Relations to Other Functions
7: William P. Reinhardt
Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. …
8: Errata
  • Equation (22.19.3)
    22.19.3 θ ( t ) = 2 am ( t E / 2 , 2 / E )

    Originally the first argument to the function am was given incorrectly as t . The correct argument is t E / 2 .

    Reported 2014-03-05 by Svante Janson.

  • 9: Mathematical Introduction
    The NIST Handbook has essentially the same objective as the Handbook of Mathematical Functions that was issued in 1964 by the National Bureau of Standards as Number 55 in the NBS Applied Mathematics Series (AMS). … As a consequence, in addition to providing more information about the special functions that were covered in AMS 55, the NIST Handbook includes several special functions that have appeared in the interim in applied mathematics, the physical sciences, and engineering, as well as in other areas. …
    10: 8.21 Generalized Sine and Cosine Integrals
    8.21.4 si ( a , z ) = z t a - 1 sin t d t , a < 1 ,
    8.21.5 ci ( a , z ) = z t a - 1 cos t d t , a < 1 ,
    8.21.18 f ( a , z ) = si ( a , z ) cos z - ci ( a , z ) sin z ,
    8.21.19 g ( a , z ) = si ( a , z ) sin z + ci ( a , z ) cos z .
    8.21.23 g ( a , z ) = 0 cos t ( t + z ) 1 - a d t .