In mathematical analysis, the Bohr–Mollerup theorem is a theorem named after the Danish mathematicians Harald Bohr and Johannes Mollerup, who proved it. The theorem characterizes the gamma function, defined for x > 0 by
as the only function f on the interval x > 0 that simultaneously has the three properties
- f is logarithmically convex.
An elegant treatment of this theorem is in Artin's book The Gamma Function, which has been reprinted by the AMS in a collection of Artin's writings.
The theorem was first published in a textbook on complex analysis, as Bohr and Mollerup thought it had already been proved.
Statement of the theorem
is the only function that satisfies with convex and also with .
Let be a function with the assumed properties established above: and is convex, and . From the fact that we can establish
The purpose of the stipulation that forces the property to duplicate the factorials of the integers so we can conclude now that if and if exists at all. Because of our relation for , if we can fully understand for then we understand for all values of .
The slope of a line connecting two points and , call it is monotonically increasing for convex functions with . Since we have stipulated is convex we know
It is evident from this last line that a function is being sandwiched between two expressions, a common analysis technique to prove various things such as the existence of a limit, or convergence. Now we recall that the function and are both monotonically increasing. Therefore if we exponentiate each term of the inequality, we will preserve the inequalities. Continuing:
The last line is a strong statement. In particular, it is true for all values of . That is is not greater than the right hand side for any choice of and likewise, is not less than the left hand side for any other choice of . Each single inequality stands alone and may be interpreted as an independent statement. Because of this fact, we are free to choose different values of for the RHS and the LHS. In particular, if we keep for the RHS and choose for the LHS and get:
Now let . The limit drives so the left side of the last inequality is driven to equal the right side. is sandwiched in between. This can only mean that is equal to . In the context of this proof this means that has the three specified properties belonging to . Also, the proof provides a specific expression for . And the final critical part of the proof is to remember that the limit of a sequence is unique. This means that for any choice of only one possible number can exist. Therefore there is no other function with all the properties assigned to . the assumptions of this theorem to
The remaining loose end is the question of proving that makes sense for all where exists. The problem is that our first double inequality
was constructed with the constraint . If, say, then the fact that is monotonically increasing would make , contradicting the inequality upon which the entire proof is constructed. But notice
which demonstrates how to bootstrap to all values of where the limit is defined.
- Hazewinkel, Michiel, ed. (2001), "Bohr–Mollerup theorem", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W., "Bohr–Mollerup Theorem", MathWorld.
- Proof of Bohr–Mollerup theorem at PlanetMath.org.
- Alternative proof of Bohr–Mollerup theorem at PlanetMath.org.
- Artin, Emil (1964). The Gamma Function. Holt, Rinehart, Winston.
- Rosen, Michael (2006). Exposition by Emil Artin: A Selection. American Mathematical Society.
- Mollerup, J., Bohr, H. (1922). Lærebog i Kompleks Analyse vol. III, Copenhagen. (Textbook in Complex Analysis)