The numerical values of these constants are approximately 1.291285997... and 0.7834305107..., respectively.
The name "sophomore's dream"[1] is in contrast to the name "freshman's dream" which is given to the incorrect[note 1] identity . The sophomore's dream has a similar too-good-to-be-true feel, but is true.
Proof
Graph of the functions y = xx (red, lower) and y = x−x (grey, upper) on the interval x ∈ (0, 1].
The proofs of the two identities are completely analogous, so only the proof of the second is presented here.
The key ingredients of the proof are:
By uniform convergence of the power series, one may interchange summation and integration to yield
To evaluate the above integrals, one may change the variable in the integral via the substitution With this substitution, the bounds of integration are transformed to giving the identityBy Euler's integral identity for the Gamma function, one has so that
Summing these (and changing indexing so it starts at n= 1 instead of n = 0) yields the formula.
Historical proof
The original proof, given in Bernoulli,[2] and presented in modernized form in Dunham,[3] differs from the one above in how the termwise integral is computed, but is otherwise the same, omitting technical details to justify steps (such as termwise integration). Rather than integrating by substitution, yielding the Gamma function (which was not yet known), Bernoulli used integration by parts to iteratively compute these terms.
The integration by parts proceeds as follows, varying the two exponents independently to obtain a recursion. An indefinite integral is computed initially, omitting the constant of integration both because this was done historically, and because it drops out when computing the definite integral.
where denotes the falling factorial; there is a finite sum because the induction stops at 0, since n is an integer.
In this case , and they are integers, so
Integrating from 0 to 1, all the terms vanish except the last term at 1,[note 2] which yields:
This is equivalent to computing Euler's integral identity for the Gamma function on a different domain (corresponding to changing variables by substitution), as Euler's identity itself can also be computed via an analogous integration by parts.
^Incorrect in general, but correct when one is working in a commutative ring of prime characteristicp with n being a power of p. The correct result in a general commutative context is given by the binomial theorem.
^All the terms vanish at 0 because by l'Hôpital's rule (Bernoulli omitted this technicality), and all but the last term vanish at 1 since log 1 = 0.
References
Formula
Bernoulli, Johann (1697). Opera omnia. Vol. 3. pp. 376–381.
Dunham, William (2005). "Chapter 3: The Bernoullis (Johann and )". The Calculus Gallery, Masterpieces from Newton to Lebesgue. Princeton University Press. pp. 46–51. ISBN 9780691095653.
OEIS, (sequence A083648 in the OEIS) and (sequence A073009 in the OEIS)
Max R. P. Grossmann (2017): Sophomore's dream. 1,000,000 digits of the first constant
Function
Literature for x^x and Sophomore's Dream, Tetration Forum, 03/02/2010
The Coupled Exponential, Jay A. Fantini, Gilbert C. Kloepfer, 1998
Sophomore's Dream Function, Jean Jacquelin, 2010, 13 pp.
Lehmer, D. H. (1985). "Numbers associated with Stirling numbers and xx". Rocky Mountain Journal of Mathematics. 15 (2): 461. doi:10.1216/RMJ-1985-15-2-461.
Gould, H. W. (1996). "A Set of Polynomials Associated with the Higher Derivatives of y = xx". Rocky Mountain Journal of Mathematics. 26 (2): 615. doi:10.1216/rmjm/1181072076.