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Difference between revisions of "Bohr-Mollerup theorem"

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The [[Gamma-function|gamma-function]] on the positive real axis is the unique positive, logarithmically convex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120330/b1203301.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120330/b1203302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120330/b1203303.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120330/b1203304.png" />.
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The [[Gamma-function|gamma-function]] on the positive real axis is the unique positive, logarithmically convex function $f$ such that $f(1)=1$ and $f(x+1) = xf(x)$ for all $x$.
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.P. Boas,   "Bohr's power series theorem in several variables"  ''Proc. Amer. Math. Soc.'' , '''125'''  (1997)  pp. 2975–2979</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Caratheodory,   "Theory of functions of a complex variable" , '''1''' , Chelsea  (1983)  pp. Sects. 274–275</TD></TR></table>
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|valign="top"|{{Ref|Ar}}||valign="top"| E. Artin, "The gamma function", Holt, Rinehart &amp; Winston (1964)
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|valign="top"|{{Ref|Bo}}||valign="top"| H.P. Boas, "Bohr's power series theorem in several variables"  ''Proc. Amer. Math. Soc.'', '''125'''  (1997)  pp. 2975–2979
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|valign="top"|{{Ref|Ca}}||valign="top"| C. Caratheodory, "Theory of functions of a complex variable", '''1''', Chelsea  (1983)  Sects. 274–275
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Latest revision as of 21:40, 27 April 2012

2020 Mathematics Subject Classification: Primary: 33B15 [MSN][ZBL]

The gamma-function on the positive real axis is the unique positive, logarithmically convex function $f$ such that $f(1)=1$ and $f(x+1) = xf(x)$ for all $x$.

References

[Ar] E. Artin, "The gamma function", Holt, Rinehart & Winston (1964)
[Bo] H.P. Boas, "Bohr's power series theorem in several variables" Proc. Amer. Math. Soc., 125 (1997) pp. 2975–2979
[Ca] C. Caratheodory, "Theory of functions of a complex variable", 1, Chelsea (1983) Sects. 274–275
How to Cite This Entry:
Bohr-Mollerup theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bohr-Mollerup_theorem&oldid=12494
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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