Given an expression involving known constants, integration in finite terms, computation of limits, etc., the constant problem is the determination of if the expression is
equal to zero. The constant problem, sometimes also called
the identity problem (Richardson 1968) is a very difficult unsolved problem in transcendental
number theory. However, it is known that the problem
is undecidable if the expression involves oscillatory
functions such as sine. However, the Ferguson-Forcade
algorithm is a practical algorithm for determining if there exist integers for given real numbers such that
or else establishing bounds within which no relation can exist (Bailey 1988).
Bailey, D. H. "Numerical Results on the Transcendence of Constants Involving ,
, and Euler's Constant." Math.
Comput.50, 275-281, 1988.Chow, T. Y. "What is
a Closed-Form Number." Amer. Math. Monthly106, 440-448, 1999.Chen,
Z.-Z. and Kao, M.-Y. "Reducing Randomness via Irrational Numbers." 7 Jul
1999. http://arxiv.org/abs/cs.DS/9907011.Richardson,
D. "Some Unsolvable Problems Involving Elementary Functions of a Real Variable."
J. Symbolic Logic33, 514-520, 1968.Richardson, D. "The
Elementary Constant Problem." In Proc. Internat. Symp. on Symbolic and Algebraic
Computation, Berkeley, July 27-29, 1992 (Ed. P. S. Wang). ACM Press,
1992.Richardson, D. "How to Recognize Zero." J. Symb. Comp.24,
627-645, 1997.Sackell, J. "Zero-Equivalence in Function Fields
Defined by Algebraic Differential Equations." Trans. Amer. Math. Soc.336,
151-171, 1993.
for given real numbers 
such that
,
, and Euler's Constant." Math.
Comput. 50, 275-281, 1988.Chow, T. Y. "What is
a Closed-Form Number." Amer. Math. Monthly 106, 440-448, 1999.Chen,
Z.-Z. and Kao, M.-Y. "Reducing Randomness via Irrational Numbers." 7 Jul
1999.