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- W85209519 startingPage "49" @default.
- W85209519 abstract "In analogy with Artin’s axiomatic treatment of the gamma function, it is here investigated to what extent Γ:IR+ → IR+ can be characterized as a convex solution of suitable combinations of the functional equations <m:math display='block'> <m:mrow> <m:mi>f</m:mi><m:mrow><m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi><m:mo>+</m:mo><m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>x</m:mi><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo></m:mrow><m:mo>,</m:mo><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo> <m:mrow> <m:mfrac> <m:mi>x</m:mi> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo>)</m:mo></m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo> <m:mrow> <m:mfrac> <m:mrow> <m:mi>x</m:mi><m:mo>+</m:mo><m:mn>1</m:mn> </m:mrow> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mn>2</m:mn><m:msqrt> <m:mi>π</m:mi> </m:msqrt> <m:msup> <m:mn>2</m:mn> <m:mrow> <m:mo>−</m:mo><m:mi>x</m:mi> </m:mrow> </m:msup> <m:mi>f</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo></m:mrow><m:mo>,</m:mo><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo></m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn><m:mo>−</m:mo><m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>π</m:mi><m:mo>/</m:mo><m:mi>sin</m:mi><m:mi>π</m:mi><m:mi>x</m:mi><m:mo>.</m:mo> </m:mrow> </m:math> $$ fleft( {x + 1} right) = xfleft( x right),fleft( {frac{x}{2}} right)fleft( {frac{{x + 1}}{2}} right) = 2sqrt pi {2^{ - x}}fleft( x right),fleft( x right)fleft( {1 - x} right) = pi /sin pi x. $$" @default.
- W85209519 created "2016-06-24" @default.
- W85209519 creator A5016438969 @default.
- W85209519 date "1978-01-01" @default.
- W85209519 modified "2023-10-17" @default.
- W85209519 title "Convexity in the Theory of the Gamma Function" @default.
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- W85209519 doi "https://doi.org/10.1007/978-3-0348-5563-1_7" @default.
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