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)abbrev package ELFUTS EllipticFunctionsUnivariateTaylorSeries
++ Author: Bill Burge, Clifton J. Williamson
++ Date Created: 1986
++ Date Last Updated: 17 February 1992
++ Description: 
++ The elliptic functions sn, sc and dn are expanded as Taylor series.

EllipticFunctionsUnivariateTaylorSeries(Coef,UTS) : SIG == CODE where
  Coef : Field
  UTS : UnivariateTaylorSeriesCategory Coef
 
  L   ==> List
  I   ==> Integer
  RN  ==> Fraction Integer
  ST  ==> Stream Coef
  STT ==> StreamTaylorSeriesOperations Coef
  YS  ==> Y$ParadoxicalCombinatorsForStreams(Coef)
 
  SIG ==> with

    sn : (UTS,Coef) -> UTS
      ++\spad{sn(x,k)} expands the elliptic function sn as a Taylor
      ++ series.

    cn : (UTS,Coef) -> UTS
      ++\spad{cn(x,k)} expands the elliptic function cn as a Taylor
      ++ series.

    dn : (UTS,Coef) -> UTS
      ++\spad{dn(x,k)} expands the elliptic function dn as a Taylor
      ++ series.

    sncndn : (ST,Coef) -> L ST
      ++\spad{sncndn(s,c)} is used internally.
 
  CODE ==> add

    import StreamTaylorSeriesOperations Coef
    UPS==> StreamTaylorSeriesOperations Coef
    integrate ==> lazyIntegrate
    sncndnre:(Coef,L ST,ST,Coef) -> L ST

    sncndnre(k,scd,dx,sign) ==
            [integrate(0,      scd.2*$UPS scd.3*$UPS dx),  _
             integrate(1,  sign*scd.1*$UPS scd.3*$UPS dx),  _
             integrate(1,sign*k**2*$UPS scd.1*$UPS scd.2*$UPS dx)]
 
    sncndn(z,k) ==
      empty? z => [0 :: ST,1 :: ST,1::ST]
      frst z = 0 => YS(x +-> sncndnre(k,x,deriv z,-1),3)
      error "ELFUTS:sncndn: constant coefficient should be 0"

    sn(x,k)  == series sncndn.(coefficients x,k).1

    cn(x,k)  == series sncndn.(coefficients x,k).2

    dn(x,k)  == series sncndn.(coefficients x,k).3