The Gamma Function Behavior Area Under the Curve Critical Points - PowerPoint PPT Presentation

The Gamma Function N. Cannady, T. Ngo, A. Williamson Introduction Motivation and History Definition Related Functions The Gamma Function Behavior Area Under the Curve Critical Points The Bluntness of The N. Cannady, T. Ngo, A. Williamson Gamma Function Conclusion Louisiana State University Bibliography SMILE REU July 9, 2010

Motivation and History The Gamma Function N. Cannady, T. Ngo, A. Williamson Introduction Motivation and History Definition Related Functions Behavior ◮ Developed as the unique extension of the factorial to Area Under the Curve non-integral values. Critical Points The Bluntness of The Gamma Function Conclusion Bibliography

Motivation and History The Gamma Function N. Cannady, T. Ngo, A. Williamson Introduction Motivation and History Definition Related Functions Behavior ◮ Developed as the unique extension of the factorial to Area Under the Curve non-integral values. Critical Points The Bluntness of The ◮ Many applications in physics, differential equations, Gamma Function statistics, and analytic number theory. Conclusion Bibliography

Motivation and History The Gamma Function N. Cannady, T. Ngo, A. Williamson Introduction Motivation and History Definition Related Functions Behavior ◮ Developed as the unique extension of the factorial to Area Under the Curve non-integral values. Critical Points The Bluntness of The ◮ Many applications in physics, differential equations, Gamma Function statistics, and analytic number theory. Conclusion ◮ ”Each generation has found something of interest to Bibliography say about the gamma function. Perhaps the next generation will also.” -Philip J. Davis

Definition The Gamma Function N. Cannady, T. Ngo, A. Williamson Introduction Motivation and History The Gamma function is an extension of the factorial Definition Related Functions (with the argument shifted down) to the complex plane. Behavior Area Under the Curve Critical Points The Bluntness of The Gamma Function Conclusion Bibliography

Definition The Gamma Function N. Cannady, T. Ngo, A. Williamson Introduction Motivation and History The Gamma function is an extension of the factorial Definition Related Functions (with the argument shifted down) to the complex plane. Behavior The basic integral definition is Area Under the Curve Critical Points � ∞ x s − 1 e − x dx . The Bluntness of The Γ( s ) = Gamma Function 0 Conclusion For the positive integers, Bibliography Γ( s ) = ( s − 1)! .

Definition The Gamma Function N. Cannady, T. Ngo, A. Williamson Introduction Motivation and History The Gamma function is an extension of the factorial Definition Related Functions (with the argument shifted down) to the complex plane. Behavior The basic integral definition is Area Under the Curve Critical Points � ∞ x s − 1 e − x dx . The Bluntness of The Γ( s ) = Gamma Function 0 Conclusion For the positive integers, Bibliography Γ( s ) = ( s − 1)! . The Gamma function is analytic for all complex numbers except the non-positive integers. The function has simple poles at these values, with residues given by ( − 1) s s ! .

Alternate Definitions and Functional Equations The Gamma Function N. Cannady, T. Ngo, A. Williamson Introduction Motivation and History Definition Related Functions Behavior Area Under the Curve A few of the most useful ones: Critical Points n s n ! 1. Γ( s ) = lim n →∞ s ( s +1) ... ( s + n ) for s � = 0 , − 1 , − 2 , ... The Bluntness of The Gamma Function n ) e − s / n ∀ s . Γ( s ) = se γ s � ∞ 1 n =1 (1 + s 2. Conclusion Bibliography 3. Γ( s + 1) = s Γ( s ). π 4. Γ( s )Γ(1 − s ) = sin π s .

The Digamma and Polygamma Functions The Gamma Function N. Cannady, T. Ngo, A. Williamson Introduction Motivation and History Definition ◮ The Digamma function, Ψ (0) ( x ) is defined as the Related Functions Behavior derivative of the logarithm of Γ( x ). Area Under the Curve Critical Points Ψ (0) ( x ) = d dx (log Γ( x )) = Γ ′ ( x ) The Bluntness of The Gamma Function Γ( x ) Conclusion Bibliography

The Digamma and Polygamma Functions The Gamma Function N. Cannady, T. Ngo, A. Williamson Introduction Motivation and History Definition ◮ The Digamma function, Ψ (0) ( x ) is defined as the Related Functions Behavior derivative of the logarithm of Γ( x ). Area Under the Curve Critical Points Ψ (0) ( x ) = d dx (log Γ( x )) = Γ ′ ( x ) The Bluntness of The Gamma Function Γ( x ) Conclusion Bibliography ◮ The Polygamma function, Ψ ( k ) ( x ) is the generalization to higher derivatives. Ψ ( k ) ( x ) = d k +1 dx k +1 (log Γ( x ))

The Euler Gamma The Gamma Function N. Cannady, T. Ngo, A. Williamson Introduction Motivation and History Definition Related Functions Behavior Area Under the Curve Critical Points ◮ The Euler Gamma arises often when discussing the The Bluntness of The Gamma function. Gamma Function Conclusion Bibliography

The Euler Gamma The Gamma Function N. Cannady, T. Ngo, A. Williamson Introduction Motivation and History Definition Related Functions Behavior Area Under the Curve Critical Points ◮ The Euler Gamma arises often when discussing the The Bluntness of The Gamma function. Gamma Function ◮ γ = lim r →∞ (log r − 1 − 1 2 − 1 3 − . . . − 1 r ) Conclusion Bibliography

The Euler Gamma The Gamma Function N. Cannady, T. Ngo, A. Williamson Introduction Motivation and History Definition Related Functions Behavior Area Under the Curve Critical Points ◮ The Euler Gamma arises often when discussing the The Bluntness of The Gamma function. Gamma Function ◮ γ = lim r →∞ (log r − 1 − 1 2 − 1 3 − . . . − 1 r ) Conclusion Bibliography ◮ It is unknown whether γ is algebraic or transcendental.

Graph of the Gamma Function The Gamma Function N. Cannady, T. Ngo, A. Williamson Introduction Motivation and History Definition Related Functions Behavior Area Under the Curve Critical Points The Bluntness of The Gamma Function Conclusion Bibliography

Overview of Behavior The Gamma Function N. Cannady, T. Ngo, A. Williamson We looked at several features of the graph: Introduction Motivation and History Definition 10 Related Functions Behavior 5 Area Under the Curve Critical Points � 4 � 2 2 4 The Bluntness of The � 5 Gamma Function Conclusion � 10 Bibliography

Overview of Behavior The Gamma Function N. Cannady, T. Ngo, A. Williamson We looked at several features of the graph: Introduction Motivation and History Definition 10 Related Functions Behavior 5 Area Under the Curve Critical Points � 4 � 2 2 4 The Bluntness of The � 5 Gamma Function Conclusion � 10 Bibliography ◮ The area under the curve.

Overview of Behavior The Gamma Function N. Cannady, T. Ngo, A. Williamson We looked at several features of the graph: Introduction Motivation and History Definition 10 Related Functions Behavior 5 Area Under the Curve Critical Points � 4 � 2 2 4 The Bluntness of The � 5 Gamma Function Conclusion � 10 Bibliography ◮ The area under the curve. ◮ Critical points of the graph for negative values shift progressively leftwards.

Overview of Behavior The Gamma Function N. Cannady, T. Ngo, A. Williamson We looked at several features of the graph: Introduction Motivation and History Definition 10 Related Functions Behavior 5 Area Under the Curve Critical Points � 4 � 2 2 4 The Bluntness of The � 5 Gamma Function Conclusion � 10 Bibliography ◮ The area under the curve. ◮ Critical points of the graph for negative values shift progressively leftwards. ◮ The graph restricted to intervals between the discontinuities looks like a squeezed segment of the graph in the positive regime.

Overview of Behavior The Gamma Function N. Cannady, T. Ngo, A. Williamson We looked at several features of the graph: Introduction Motivation and History Definition 10 Related Functions Behavior 5 Area Under the Curve Critical Points � 4 � 2 2 4 The Bluntness of The � 5 Gamma Function Conclusion � 10 Bibliography ◮ The area under the curve. ◮ Critical points of the graph for negative values shift progressively leftwards. ◮ The graph restricted to intervals between the discontinuities looks like a squeezed segment of the graph in the positive regime. ◮ Critical points for negative values approach zero.

Questions About the Integral of Γ( x ) The Gamma Function N. Cannady, T. Ngo, A. Williamson When considering the graph of the Gamma Function, one Introduction might be lead to consider several things. Motivation and History Definition Related Functions Behavior 10 Area Under the Curve Critical Points 5 The Bluntness of The Gamma Function � 4 � 2 2 4 Conclusion � 5 Bibliography � 10

Questions About the Integral of Γ( x ) The Gamma Function N. Cannady, T. Ngo, A. Williamson When considering the graph of the Gamma Function, one Introduction might be lead to consider several things. Motivation and History Definition Related Functions Behavior 10 Area Under the Curve Critical Points 5 The Bluntness of The Gamma Function � 4 � 2 2 4 Conclusion � 5 Bibliography � 10 ◮ Does the integral of Γ( x ) converge if one of the bounds of integration is a point of discontinuity?