Intereting Posts

Laplace equation Fourier transform
find the measure of $AMC$
Translation-invariant metric on locally compact group
Matrix on complex field.
proving the inequality $\triangle\leq \frac{1}{4}\sqrt{(a+b+c)\cdot abc}$
How to solve in radicals this family of equations for any degree $k$?
How to factor ideals in a quadratic number field?
What is wrong with the sum of these two series?
Show that exist $i>0$ such that the Fibonacci number $F_{i}$ is divisible by 2015
Subspace generated by permutations of a vector in a vector space
Show $\lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n} = \|f\|_{\infty}$ for $f \in L^{\infty}$
Splitting of conjugacy class in alternating group
Suppose $f: \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$
Conditional expectation and almost sure equality
solve$\frac{xdx+ydy}{xdy-ydx}=\sqrt{\frac{a^2-x^2-y^2}{x^2+y^2}}$

It always puzzles me, how the Gamma function’s inventor came up with its definition

$$\Gamma(x+1)=\int_0^1(-\ln t)^x\;\mathrm dt=\int_0^\infty t^xe^{-t}\;\mathrm dt$$

Is there a nice derivation of this generalization of the factorial?

- Geometric meaning of the determinant of a matrix
- What is an intuitive explanation for $\operatorname{div} \operatorname{curl} F = 0$?
- A series problem by Knuth
- Geometric justification for the prime spectrum and “generic points”
- Intuitive Proof of the Chain Rule in 1 Variable
- In every set of $14$ integers there are two that their difference is divisible by $13$
- What application is there for a non-Hausdorff topological space?
- Showing that $10!=6!7!$ via Gamma function
- The Intuition behind l'Hopitals Rule
- Proving and deriving a Gamma function

Here is a nice paper of Detlef Gronau Why is the gamma function

so as it is?.

Concerning alternative possible definitions see Is the Gamma function mis-defined? providing another resume of the story Interpolating the natural factorial n! .

Concerning Euler’s work Ed Sandifer’s articles ‘How Euler did it’ are of value too, in this case ‘Gamma the function’.

I guess you can say this is yet another application of the power of integration by parts (and I am guessing that is how the integral formula “was come up with” initially).

If you are trying to find the antiderivative of $P(t) e^t$, where $P(t)$ is a polynomial, integration by parts arises naturally and I would say it(integral of $P(t) e^t$) is quite natural to encounter during ones study of mathematics. And if you actually work it out, you notice the factorial like recursion. We can rid of the “non-integral” parts of the integration by parts formula by using the limits $0$ and $\infty$.

If $I_n = \int_{0}^{\infty} t^n e^{-t} \text{dt}$ then integration by parts gives us

$$I_n = -e^{-t}t^n|_0^{\infty} + n\int_{0}^{\infty} t^{n-1} e^{-t} = nI_{n-1}$$

so if

$f(x) = \int_{0}^{\infty} t^x e^{-t} \text{dt}, \quad x \ge 0$

then

$f(x) = x f(x-1), \quad x \ge 1$.

Also, we have that $f(0) = 1$, thus the integral definition agrees with the factorial function at the non-negative integers and can serve as a real extension for factorial.

Using Analytic continuation its domain can be extended further.

$$

\int e^{ax} dx = \frac{1}{a} e^{ax} + c

$$

Take $\left .\frac{d}{da}\right |_{a=1}$ on both sides $n$ times, and algebra to get rid of $(-1)^n$, you’ll have an integral equal to $n!$.

This is an intuitive way to get the Gamma function. You’ve shown that for integers it holds from this simple derivation.

Mathematicians then went through a great deal of work to show that it holds true for allot more than just the integer case.

- REVISITED $^2$: Does a solution in $\mathbb{R}^n$ imply a solution in $\mathbb{Q}^n$?
- A question on morphisms of fields
- Prove that if $\sum_{n=1}^\infty a_{2n}$ and $\sum_{n=1}^\infty a_{2n-1}$ converge then $\sum_{n=1}^\infty a_n$ converges.
- Criterion for locally free modules of rank $1$
- sum of arctangent
- Finding the maximum value of a function on an ellipse
- How to prove that the normalizer of diagonal matrices in $GL_n$ is the subgroup of generalized permutation matrices?
- Multiplicative norm on $\mathbb{R}$.
- Combination of quadratic and arithmetic series
- Why does the semigroup commute with integration?
- $(\sin^{-1} x)+ (\cos^{-1} x)^3$
- The Stone-Čech compactification of a space by the maximal ideals of the ring of bounded continuous functions from the space to $\mathbb{R}$
- Completing an exact sequence
- A closed form for the integral $\int_0^1\frac{1}{\sqrt{y^3(1-y)}}\exp\left(\frac{i A}{y}+\frac{i B}{1-y}\right)dy$
- Is this Fourier like transform equal to the Riemann zeta function?