Monday Math 18: Gamma Function (Part 1/?)

The gamma function is defined for all complex z except at z=0,-1,-2,-3,…, by
We see first that . Next, we can apply integration by parts to for real x to get:
.
Thus for an integer n, we see
Thus the gamma function is an extension of the factorial to non-integer and non-real values.
We can also compute :
.
Making the u-substitution , we get:
.
And so


and, thus for integer n:

We also have


and so on.

If we take the original integral and make the substitution , we get:


And thus for an integer n>0,


Also, according to the Bohr-Mollerup theorem, the gamma function is the only function on x>0 for which the following three properties hold:
I.
II.
III. is convex.

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9 Responses to “Monday Math 18: Gamma Function (Part 1/?)”

  1. Monday Math 19: The Gamma Function (Part 2/?) « Twisted One 151’s Weblog Says:

    […] The Gamma Function Part 2: Weierstrass form and Stirling’s formulas (Part 1 is here) Let us now define a sequence of functions . We note first that has simple poles at 0, -1, -2, […]

  2. Monday Math 20: The Gamma Function (Part 3/?) « Twisted One 151’s Weblog Says:

    […] Gamma Function (Part 3/?) The Gamma Function Part 3: Beta Function and Legendre duplication: (Part 1) (Part 2) Consider the function defined by . The integral converges for x,y>0; and the […]

  3. Monday Math 21: The Gamma Function (Part 4/?) « Twisted One 151’s Weblog Says:

    […] (Part 4/?) The Gamma Function Part 4: Euler Reflection Formula and the Sine Function: (Part 1) (Part 2) (Part 3) Let us now define the function , for non-integer x, as . Now, we remember that […]

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    […] Math 22: The Gamma Function (Part 5/5) Part 5: n-ball (Part 1) (Part 2) (Part 3) (Part 4) In mathematics, a ball is the region of three dimensional space […]

  5. Monday Math 24: Repeated Integrals « Twisted One 151’s Weblog Says:

    […] n>0 even if n is not an integer. Recall, however, that for positive integer n, , where is the gamma function. Thus, if we replace the factorial in our formula with the corresponding gamma function, we obtain […]

  6. Monday Math 29 « Twisted One 151’s Weblog Says:

    […] . Then and , so our integral becomes: , and you should recognize that last integral as the gamma function of x. Therefore, we find: , and that last sum is also familiar: it is the Riemann zeta function. […]

  7. Monday Math 41: The Digamma Function « Twisted One 151’s Weblog Says:

    […] Math 41: The Digamma Function Finding the derivative of the gamma function is not an easy task. However, there are reasons to do so. In particular, mathematicians define the […]

  8. Monday Math 103 « Twisted One 151's Weblog Says:

    […] region of convergence. Using the u-substitution u=st, . You should recognise this last integral as the gamma function of r+1, and so , and so we see that it is necessary that r not be a negative integer. If r is a […]

  9. Gonzalo Pettet Says:

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