Intereting Posts

Find the equation of the tangent line to $y=x^4-4x^3-5x+7$
Generating functions and the Riemann Zeta Function
What is “algebra” in $\sigma$-algebra (or “field” in $\sigma$-field)?
How to find a matrix square root with all real entries (if it exists)
Show A is countable infinity
Distributional derivative of a hölder function
Find the number of simple labeled graphs which have no isolated vertices
Difference between “≈”, “≃”, and “≅”
how to solve double integral of a min function
inductive proof for $\binom{2n}{n}$
How to show an infinite number of algebraic numbers $\alpha$ and $\beta$ for $_2F_1\left(\frac14,\frac14;\frac34;-\alpha\right)=\beta\,$?
application of strong vs weak law of large numbers
Probability of Drawing a Card from a Deck
Jobs in industry for pure mathematicians
Computation with a memory wiped computer

I Once heard on the internet that $(\frac{1}{2})!=\frac{\sqrt{\pi}}{2}$ so now, I’m wondering what $(\frac{1}{4})!$ is equal to?

My attempt:

Since $(\frac{1}{2})!=\frac{\sqrt{\pi}}{2}$ and since $\frac{1}{4}=\frac{1}{2}\div2$ then:

$$\begin{align}

(\frac{1}{4})!=\frac{\sqrt{\pi}}{2}\div2= \\

\frac{\sqrt{\pi}}{4}

\end{align}$$

Is my assumption correct? If not, what is the true answer?

- Can n! be a perfect square when n is an integer greater than 1?
- Why is this formula for $(2m-1)!!$ correct?
- Number of solutions for $\frac{1}{X} + \frac{1}{Y} = \frac{1}{N!}$ where $1 \leq N \leq 10^6$
- Closed Form for Factorial Sum
- Prove by induction that $n!>2^n$
- An identity involving the Pochhammer symbol

By the way, I asked this question just because I am curious.

- Prove $\binom{2p+1}{p}\equiv2$ mod $p$ when $p$ is any prime.
- Sum of $\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$
- Euler's identity in matrix form
- Find $n$, where its factorial is a product of factorials
- Evaluating $\int_{0}^{\infty} \left ^{-1}\mathrm{d}x$
- Puzzle on the triangle.
- What are the symmetries of a colored rubiks cube?
- Simplify : $( \sqrt 5 + \sqrt6 + \sqrt7)(− \sqrt5 + \sqrt6 + \sqrt7)(\sqrt5 − \sqrt6 + \sqrt7)(\sqrt5 + \sqrt6 − \sqrt7) $
- How to prove that $\lim_{n \to\infty} \frac{(2n-1)!!}{(2n)!!}=0$
- Integer factorization: What is the meaning of $d^2 - kc = e^2$

Your logic is incorrect. That is not how the factorial nor the Gamma function behave. Take for example $4!$. Since $2=4\div2$, you seem to think that $2!=4!/2$, but a quick check says this is wrong.

Thanks to expressions for values of the gamma function and Wikipedia, it is known that

$$(1/4)!=\Gamma(5/4)=\frac12\pi^{1/4}K\left(\frac1{\sqrt2}\right)^{1/2}\approx0.90640247705$$

where $K(x)$ is the elliptic $K$ function (complete elliptic integral of the first kind).

In general, for non-integer $x$, we usually extend the factorial as follows:

$$x!=\int_0^\infty t^xe^{-t}\ dt$$

for $x>-1$. Other forms may be given in the first link, and for your specific problem, many forms are given in the Wikipdia.

**From here**:

We have $$(1/4)! =\Gamma (5/4) =\Gamma (1/4)\frac {(4 (1)-3)!!!!}{4^1} =\Gamma (1/4)\frac {1}{4} \approx 0.90640$$

Also in general for any $x$, $$(x)! \neq \frac {(2x)!}{2} $$ Hope it helps.

- show that $\frac{a^2+b^2+c^2}{15}$ is non-square integer
- Why does zeta have infinitely many zeros in the critical strip?
- Integral equation solution: $y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$
- Prove that this function is bounded
- How many different ways can you distribute 5 apples and 8 oranges among six children?
- What does “removing a point” have to do with homeomorphisms?
- Non-linear function on $\mathbb{R}^2$ preserving the origin and maps lines onto lines?
- Sum and Product of two transcendental numbers cannot be simultaneously algebraic
- Analogies between finite groups and Lie groups
- Support vs range of a random variable
- Who are some blind or otherwise disabled mathematicians who have made important contributions to mathematics?
- Rolle's Theorem and the Mean Value Theorem
- Projective Nullstellensatz
- Structure of ideals in the product of two rings
- Is there an algorithm for writing an integer as a difference of squares?