Sophomore's dream explained

In mathematics, the sophomore's dream is the pair of identities (especially the first)

discovered in 1697 by Johann Bernoulli.

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^[2] identity The sophomore's dream has a similar too-good-to-be-true feel, but is true.

Proof

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:

• to write $x^x\; =\; \backslash exp(x\backslash ln\; x)$ (using the notation for the natural logarithm and for the exponential function);
• to expand $\backslash exp(x\backslash ln\; x)$ using the power series for ; and
• to integrate termwise, using integration by substitution.

In details, can be expanded as

$$x^x\; =\; \backslash exp(x\; \backslash log\; x)\; =\; \backslash sum\_^\backslash infty\; \backslash frac.$$

Therefore,

$$\backslash int\_0^1\; x^x\backslash ,dx\; =\; \backslash int\_0^1\; \backslash sum\_^\backslash infty\; \backslash frac\; \backslash ,dx.$$

By uniform convergence of the power series, one may interchange summation and integration to yield

$$\backslash int\_0^1\; x^x\backslash ,dx\; =\; \backslash sum\_^\backslash infty\; \backslash int\_0^1\; \backslash frac\; \backslash ,dx.$$

To evaluate the above integrals, one may change the variable in the integral via the substitution $x=\backslash exp(-\backslash frac).$ With this substitution, the bounds of integration are transformed to

0 < u < inf ty, giving the identity$$\backslash int\_0^1\; x^n(\backslash log\; x)^n\backslash ,dx\; =\; (-1)^n\; (n+1)^\; \backslash int\_0^\backslash infty\; u^n\; e^\backslash ,du.$$By Euler's integral identity for the Gamma function, one has $$\backslash int\_0^\backslash infty\; u^n\; e^\backslash ,du=n!,$$so that$$\backslash int\_0^1\; \backslash frac\backslash ,dx=\; (-1)^n\; (n+1)^.$$

Summing these (and changing indexing so it starts at instead of) yields the formula.

Historical proof

The original proof, given in Bernoulli, and presented in modernized form in Dunham, differs from the one above in how the termwise integral $\backslash int\_0^1\; x^n(\backslash log\; x)^n\backslash ,dx$ 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.

+ C both because this was done historically, and because it drops out when computing the definite integral.

Integrating $\backslash int\; x^m\; (\backslash log\; x)^n\backslash ,dx$ by substituting $u\; =\; (\backslash log\; x)^n$ and $dv=x^m\backslash ,dx$ yields:

(also in the list of integrals of logarithmic functions). This reduces the power on the logarithm in the integrand by 1 (from

n to n -1 ) and thus one can compute the integral inductively, as$$\backslash int\; x^m\; (\backslash log\; x)^n\backslash ,dx=\; \backslash frac\; \backslash cdot\; \backslash sum\_^n\; (-1)^i\; \backslash frac\; (\backslash log\; x)^$$

where $(n)\_i$ denotes the falling factorial; there is a finite sum because the induction stops at 0, since is an integer.

In this case $m=n$, and they are integers, so

$$\backslash int\; x^n\; (\backslash log\; x)^n\backslash ,dx=\; \backslash frac\; \backslash cdot\; \backslash sum\_^n\; (-1)^i\; \backslash frac\; (\backslash log\; x)^.$$

Integrating from 0 to 1, all the terms vanish except the last term at 1,^[3] which yields:

$$\backslash int\_0^1\; \backslash frac\backslash ,dx=\; \backslash frac\backslash frac\; (-1)^n\; \backslash frac\; =\; (-1)^n\; (n+1)^.$$

This is equivalent to computing Euler's integral identity

\Gamma(n +1 ) = n! 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.

See also

• Series (mathematics)

References

Formula

• Book: Bernoulli, Johann . 1697 . Opera omnia . 3 . 376–381.
• Book: Borwein . Jonathan . Jonathan Borwein . Bailey . David H. . David H. Bailey (mathematician) . Girgensohn . Roland . 2004 . Experimentation in Mathematics: Computational Paths to Discovery . 4, 44 . 9781568811369.
• Book: Dunham, William . 2005 . Chapter 3: The Bernoullis (Johann and x ^{x} ) . The Calculus Gallery, Masterpieces from Newton to Lebesgue . Princeton University Press . 46–51 . 9780691095653.
• OEIS, and
  • • 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. . 10.1216/RMJ-1985-15-2-461. Numbers associated with Stirling numbers and x^x . Rocky Mountain Journal of Mathematics . 15 . 461 . 1985 . 2 . free .
• Gould . H. W. . 10.1216/rmjm/1181072076 . A Set of Polynomials Associated with the Higher Derivatives of y = x^x. Rocky Mountain Journal of Mathematics . 26 . 615 . 1996 . 2 . free .

Footnotes

Notes and References

1. It appears in .
2. Incorrect in general, but correct when one is working in a commutative ring of prime characteristic with being a power of . The correct result in a general commutative context is given by the binomial theorem.
3. All the terms vanish at 0 because $\backslash lim\_\; x^m\; (\backslash log\; x)^n\; =\; 0$ by l'Hôpital's rule (Bernoulli omitted this technicality), and all but the last term vanish at 1 since .