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AngouriMath.Functions.Algebra

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Classes within the AngouriMath.Functions.Algebra namespace

• AsymptoticSeries

  Summary

  A power series in w around w -> 0+, truncated at a known exponent, whose
  coefficients are arbitrary expressions not containing w.
  

  Remarks

  This is what Gruntz's algorithm expands in, and it is not a Taylor series: the
  exponents may be negative and fractional, and the coefficients are symbolic rather
  than numeric. What is wanted from it in the end is only the leading term, but the
  leading terms of a sum can cancel -- which is the whole point of the algorithm, since
  e^(x + e^-x) - e^x cancels to every order when its two parts are expanded separately
  -- so the series has to be carried far enough to see past the cancellation, and how
  far that is is not known in advance. The caller raises the order until a leading term
  survives.
  The value of log(w) is supplied from outside rather than left as a logarithm of
  a symbol. Gruntz chooses w as an exponential, so its logarithm is an ordinary
  expression in x, and expanding a logarithm needs it: without it the constant term of
  log(c * w^e) cannot be told from zero.
  

• EquationSolver

• Gruntz

  Summary

  Gruntz's algorithm for the limit of an expression as x tends to positive infinity,
  after D. Gruntz, "On Computing Limits in a Symbolic Manipulation System", ETH 1996.
  

  Remarks

  The idea is to work out which subexpressions grow fastest, rewrite the expression in
  terms of a single one of them, and read the answer off the leading term of the power
  series that leaves. Two functions are in the same comparability class when
  log|f| / log|g| tends to something finite and non-zero, and the set of subexpressions
  in the fastest class present is the mrv set. Every member of it can be written as a
  power of any other member times something slower, so one member is picked -- turned
  upside down if need be, so that it tends to zero -- and everything is written in terms
  of it. What is left is a series in that one quantity, and the sign of the leading
  exponent says whether the limit is zero, infinite, or the limit of the leading
  coefficient, which is a smaller problem of the same kind.
  This answers the limits that defeat term-by-term expansion because the terms cancel to
  every order. lim x -> +oo e^(x + e^(-x)) - e^x is the standard one: expanding the two
  exponentials separately gives two divergent series whose difference cancels entirely,
  while rewriting the whole expression in w = e^(-x) gives (e^w - 1)/w, whose leading
  term is 1.
  The scope here is the exp-log functions: what can be built from x and the rationals
  with the four operations, exp and log. That is the class the algorithm is proved for,
  because those functions are eventually monotone and so comparable at all. Anything
  else -- a sine, an unknown function, a factorial -- makes this decline rather than
  guess, since sin(x) has no limit at infinity and no comparability class either.
  

• IndefiniteIntegralSolver

• IntegralPatterns

• Integration

• LimitFunctional

• LimitSolvers