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Notes are from D. R. Wilton, Dept. of ECE
Fall 2016 David R. Jackson Notes 13 The Gamma Function Notes are from D. R. Wilton, Dept. of ECE
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The Gamma Function Definition # 1
The Gamma function appears in many expressions, including asymptotic series. It generalizes the factorial function n! to non-integer values and even complex values. Definition # 1 This definition gives the Gamma function a nice property, as shown on the next slide.
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The Gamma Function Factorial property: Hence or
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The Gamma Function (cont.)
Definition # 2 This is the Euler-integral form of the definition. Note: Leonard Euler Note: Definition 1 is the analytic continuation of definition 2 into the entire complex plane (except at the negative integers).
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The Gamma Function (cont.)
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The Gamma Function (cont.)
Equivalence of definitions #1 and #2
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The Gamma Function (cont.)
Definition # 3 The Weierstrass product form can be shown to be equivalent to definitions #1 and #1.
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The Gamma Function (cont.)
Euler Reflection Formula Note: If we can calculate (z) for Re(z) > 0, we can use this formula to find (z) for Re(z) < 0. The two points are reflections about the x = 1/2 line.
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The Gamma Function (cont.)
A special result that occurs frequently is (1/2). To calculate this, use the reflection formula: Set z = 1/2:
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The Gamma Function (cont.)
Pole Behavior Use
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The Gamma Function (cont.)
Residues at Poles Use Hence In general:
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The Gamma Function (cont.)
Note: There are simple poles at z = 0, -1, -2,…
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The Gamma Function (cont.)
(x) and 1 / (x) Note: (x) never goes to zero. In fact, (z) never goes to zero (1 / (z) is analytic everywhere).
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The Gamma Function (cont.)
Sterling’s formula (asymptotic series for large argument): Taking the ln of both sides, we also have Valid for
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The Gamma Function (cont.)
Summary of Factorial Generalization Integers Real numbers Complex numbers
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The Gamma Function (cont.)
Summary of Factorial Generalization Complex numbers Complex numbers
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