What is the gamma function?

The factorial is one of the simplest operations in mathematics. Take a positive whole number and multiply it by every positive whole number below it.

(The exclamation mark means “factorial”. So $5!$ is read aloud as “five factorial”.)

For whole numbers, everything seems straightforward:

And then someone asks a rather awkward question. What is

Half of a factorial? At first the question can sound almost meaningless. Factorials seem to describe a process of counting down through whole numbers. There is no obvious instruction telling us what to multiply when the input is one-half. Yet the question is not meaningless. Its answer is

A sequence built from multiplication of whole numbers has somehow acquired a value involving $\pi$. The function that makes this possible is called the gamma function. It is written

($\Gamma$ is the capital Greek letter gamma, pronounced “GAM-uh”. The lowercase Greek letter $\gamma$, used elsewhere in mathematics, denotes a different object.)

The gamma function extends the factorial pattern beyond the whole numbers. It reaches fractional numbers, real numbers and, after analytic continuation, almost the whole complex plane. But describing Gamma simply as “factorials between the factorials” misses most of the story. Gamma emerged from an eighteenth-century problem about how mathematical patterns should be continued. Its discovery involved several of the most important mathematicians of the period. Its defining integral became a model example of how apparently discrete mathematics can be transformed into continuous mathematics. Its later study helped shape complex analysis and the theory of special functions. And more than a century after Euler discovered it, Gamma appeared inside the machinery that Bernhard Riemann used to reveal the hidden symmetry of the zeta function. So to understand why

appears inside the Riemann xi function, we need to go back considerably further than Riemann. We need to begin with a sequence of numbers.

Factorials are naturally discrete

The factorial numbers grow very quickly:

INPUT       0         1         2         3         4         5
            |         |         |         |         |         |
VALUE       1         1         2         6        24       120
            *---------*---------*---------*---------*---------*
                 ?         ?         ?         ?
                       WHAT GOES HERE?

For a positive integer $n$,

But there is another way to describe exactly the same pattern. Each new factorial is obtained from the previous one by multiplying by the new number:

That relationship is more important than it first appears. Suppose we could find a function $F(x)$ satisfying

and

At the positive integers it would automatically produce $F(n)=(n-1)!$. The question becomes: can we find a genuine function obeying this rule even when $x$ is not a whole number? That question is several centuries old.

The problem began before Euler

In the seventeenth century, mathematicians were becoming increasingly comfortable with infinite series, products, interpolation and the new techniques that would eventually become calculus. John Wallis had already investigated a version of the factorial-interpolation problem in his 1655 Arithmetica Infinitorum. He obtained half-integer information, but not the general function that we now use. By the late 1720s, Daniel Bernoulli and Christian Goldbach were also thinking about how a sequence known at integer positions might be continued between them. James Stirling was working independently on closely related questions, with a strong interest in efficient numerical approximation. [1] This was the mathematical atmosphere in which the young Leonhard Euler encountered the problem. A formula such as $x^2$ already knows what to do between the integer squares: nothing prevents us from evaluating $(3/2)^2$. The factorial sequence was much more stubborn. Its values were $1,2,6,24,120,\ldots$, but there was no obvious algebraic expression waiting behind them. Euler found one of the decisive ways through.

Euler's first answer was not a curve

The interpolation problem does not ask us merely to draw something smooth through a collection of dots. Infinitely many smooth curves could do that. Euler instead sought formulas whose algebra itself reproduced the factorial behaviour. One classical limit form of the answer is

(For each finite $k$, the expression is built from ordinary multiplication and factorials. As $k$ grows without bound, the expression approaches the gamma function.)

This already gives the interpolation problem a very different character. We are not inventing values independently between the integers. We are building one limiting object that automatically agrees with the factorial pattern. Euler's work also produced the integral representation that became the most familiar doorway into Gamma. [2]

Euler, Goldbach and a problem worth writing about

Euler discussed the interpolation problem in correspondence with Christian Goldbach in 1729\. One of his earliest papers, De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt — On transcendental progressions whose general terms cannot be given algebraically — developed the subject in detail. The paper was written in 1729 and published in 1738\. [3] Euler did not write $\Gamma(x)$. That notation did not yet exist, and the modern factorial symbol was not yet his language either. He used a bracket notation $[x]$ and arrived at an integral that can be written as

With the substitution $t=e^{-u}$, this becomes

In modern Gamma notation, Euler's $[x]$ is $\Gamma(x+1)$. The one-place shift is important; it is the reason modern Gamma satisfies $\Gamma(n)=(n-1)!$ rather than $\Gamma(n)=n!$. The strange factorial problem had turned into an integral. That transformation is the heart of the story.

What actually is the gamma function?

In modern notation, for inputs with positive real part, the gamma function is defined by Euler's integral

[2]

(An integral adds up infinitely many tiny contributions. Here each positive value of $t$ contributes $t^{s-1}e^{-t}$, and Gamma is the total obtained by adding those contributions from $t=0$ all the way to infinity.)

At first sight, this looks even less like a factorial than the original question did. There are no descending whole numbers. There is an exponential. There is an integral extending to infinity. There is an unfamiliar power $t^{s-1}$. So why should this have anything to do with $1,1,2,6,24,120,\ldots$? The answer comes from one calculation.

Why Euler's integral behaves like a factorial

Consider

We can evaluate this using integration by parts. Take $u=t^s$ and $dv=e^{-t}\,dt$. Then $du=s t^{s-1}\,dt$ and $v=-e^{-t}$. So

For the values of $s$ where this integral initially applies, the boundary term vanishes. What remains is

(Move one step to the right in Gamma and the value is multiplied by the current input. That is exactly the same recursive behaviour as the factorials.)

This equation is the key. Once we know $\Gamma(1)=1$, the recurrence gives $\Gamma(2)=1$, $\Gamma(3)=2$, $\Gamma(4)=6$, and so on. Therefore

for positive integers $n$. Gamma has reproduced the factorials. But unlike the ordinary factorial operation, the integral makes sense between the integers as well.

                 WHOLE-NUMBER LADDER

 Gamma(1) --x1--> Gamma(2) --x2--> Gamma(3) --x3--> Gamma(4)
    1                1                2                6
                                                        |
                                                       x4
                                                        v
                                                    Gamma(5)
                                                       24

                  HALF-STEP LADDER

 Gamma(1/2) --x1/2--> Gamma(3/2) --x3/2--> Gamma(5/2)
   sqrt(pi)              sqrt(pi)/2          3sqrt(pi)/4

Why the numbering seems to be one step out

There is an irritating-looking detail. We might have expected $\Gamma(n)=n!$. Instead,

So $\Gamma(5)=4!=24$, not $5!=120$. If we want the direct extension of $x!$, we write

(Gamma is shifted by one place relative to the modern factorial notation. The shift makes the recurrence $\Gamma(s+1)=s\Gamma(s)$ particularly clean.)

The name and notation came later. Adrien-Marie Legendre introduced the symbol $\Gamma$ for the function in 1811\. Gauss later used a different $\Pi$ notation for a closely shifted version of the same function. [1]

Half a factorial

We can now return to our original question. What is

By the Gamma convention,

The recurrence tells us

So everything depends on $\Gamma(1/2)$. From the defining integral,

Let $t=u^2$. Then $dt=2u\,du$ and $t^{-1/2}=1/u$. The factors of $u$ cancel, leaving

The Gaussian integral satisfies

Because the integrand is symmetric,

Therefore

and so

(The appearance of $\pi$ is not a numerical coincidence. Gamma at one-half turns into the Gaussian integral, and the Gaussian integral evaluates to $\sqrt{\pi}$.)

This is one of those moments where the borders between apparently different parts of mathematics begin to disappear. A question about the sequence $1,2,6,24,120,\ldots$ has led us to an exponential curve, an infinite integral and $\pi$.

But couldn't we draw lots of curves through the factorials?

Yes. And this is an important objection. Suppose all we require is a smooth curve passing through the factorial values. There are infinitely many ways to draw one. Even requiring

does not by itself force an arbitrary positive function to be Gamma. So what makes Euler's continuation the natural factorial function? This question exposes an interesting difference between discovery and later mathematical understanding. Euler had found an extraordinarily productive continuation and derived powerful representations of it. But the modern characterization of exactly why Gamma is singled out came much later.

Nearly two centuries later: why Gamma is the natural extension

In 1922, Harald Bohr and Johannes Mollerup gave the characterization now known as the Bohr-Mollerup theorem. On the positive real numbers, Gamma is the unique positive function satisfying

and the requirement that $\log f(x)$ is convex. [2]

(Logarithmically convex means that $\log f(x)$ bends upwards in the mathematical sense of convexity. It is a strong regularity condition that rules out artificial oscillations between successive factorial values.)

This is an unexpectedly satisfying conclusion. We are not saying merely that Euler happened to choose a pleasant curve through the factorial numbers. We can say that if we demand the factorial recurrence, the correct starting value, positivity and a natural convexity condition, exactly one function works on the positive real axis. That function is Gamma.

Gamma is not merely an interpolating curve

Even the phrase “interpolating factorials” can now become misleading. Gamma does pass through the factorial values. But it is not constructed by drawing a curve through a collection of dots and smoothing the joins. It exists as an analytic object in its own right. Its integral produces the factorial values because the integral obeys the factorial recurrence. The factorials are therefore particular values of Gamma. From the modern viewpoint, we can almost turn the historical description around. Instead of saying “Gamma is a way of extending factorials”, we can say: factorials are the integer values of a much larger function. That change of perspective is very similar to what happens with the Riemann zeta function. A familiar discrete pattern turns out to be a small visible part of a much larger analytic object.

What is the exponential doing there?

There is still a reasonable question. Why does Euler's integral contain $e^{-t}$? The two pieces $t^{s-1}$ and $e^{-t}$ play very different roles. For positive $t$, the power $t^{s-1}$ responds strongly to the value of $s$. Changing $s$ changes which parts of the positive $t$-axis receive the most weight. The exponential $e^{-t}$ falls rapidly towards zero as $t$ becomes large. That rapid decay prevents the distant tail of the integral from overwhelming the total.

        t^(s-1)                    e^(-t)
           |                          |
           | changes with s           | kills the far tail
           v                          v
      +------------------------------------+
      |       t^(s-1) * e^(-t)            |
      +------------------------------------+
                       |
                       | integrate from 0 to infinity
                       v
                    Gamma(s)

Near zero, the power controls whether the integral behaves properly. Far away, the exponential wins against any fixed power and pushes the integrand rapidly towards zero. For

these effects make Euler's integral converge.

($\operatorname{Re}(s)$ means the real part of the possibly complex number $s$. If $s$ is an ordinary positive real number, the condition simply says $s>0$.)

But the Gamma function does not stop there.

Gamma enters the complex plane

Euler's integral directly defines Gamma when $\operatorname{Re}(s)>0$. The recurrence gives us a way to move left:

Repeatedly applying the same relation extends Gamma into regions where the original integral no longer converges directly. More generally, this extension is understood through analytic continuation. There are, however, some places where the recurrence forces a problem. At $s=0$ the denominator vanishes. Moving another step gives trouble at $s=-1$, then $s=-2$, and so on. Gamma therefore has simple poles at

and no zeros anywhere in the complex plane. [2]

(A meromorphic function is allowed to have isolated poles: points where its value blows up in a controlled way. Away from those poles, Gamma is analytic.)

 <------------------------------------------------------------>
   -5      -4      -3      -2      -1       0       1       2
    ^       ^       ^       ^       ^       ^       |       |
   POLE    POLE    POLE    POLE    POLE    POLE   finite  finite

So Gamma has travelled a remarkable distance from the original factorial problem. What began as $1,2,6,24,\ldots$ has become a meromorphic function of a complex variable.

A beautiful reflection

Gamma also satisfies a striking identity:

This is Euler's reflection formula. The formula relates the value of Gamma at $s$ to its value at the reflected point $1-s$. Notice the structure $s\longleftrightarrow1-s$. Readers of What is the Riemann xi function? may already recognise that reflection. It is not yet the Riemann functional equation, but Gamma has exactly the kind of reflection machinery that will later become important there. And once again, $\pi$ has appeared.

What did other mathematicians do with Euler's discovery?

The surviving history is less like a public contest over one proposed function and more like a long mathematical conversation. Daniel Bernoulli and Goldbach had already been engaged with interpolation questions. Stirling was working independently on closely related factorial problems, with a stronger emphasis on effective approximation and calculation. Euler was especially interested in analytic expressions and identities. In the mid-1730s Euler and Stirling became aware of one another's work and corresponded. [1] Legendre later organised Eulerian integrals systematically and introduced the $\Gamma$ notation. Gauss developed another treatment and a shifted $\Pi$ notation. Weierstrass later represented the reciprocal Gamma function through an infinite product. Bohr and Mollerup eventually supplied the uniqueness characterization that explains why Gamma is distinguished among functions obeying the same recurrence. By the twentieth century, mathematicians could begin from several apparently different constructions — integral, limit, infinite product or functional characterization — and prove that they describe the same object. That is a sign of mathematical maturity: a function discovered to solve one problem has acquired an entire network of equivalent descriptions.

Was Gamma challenged?

Not in quite the way a conjecture is challenged. The gamma function is a defined mathematical object. There is no proposition called “Gamma” that can be defeated by finding a counterexample. The real mathematical questions were different. Does Euler's construction reproduce the factorials? Does the integral converge? Does the recurrence hold? Can the function be continued consistently? What happens at zero and the negative integers? Are the integral, limit and product constructions genuinely equivalent? Is Euler's continuation uniquely distinguished by natural conditions? Those questions can be proved. The recurrence follows directly from integration by parts. The half-integer value follows from the Gaussian integral. Analytic continuation and functional identities expose the complex structure. Bohr-Mollerup answers the uniqueness question on the positive real axis. So Gamma did not become fundamental because mathematicians simply agreed that Euler's interpolation looked plausible. Its properties became part of a rigorous mathematical theory.

The factorials keep growing

Gamma also gives a natural way to discuss the growth of factorials. For large positive $x$,

At integer $x=n$, this becomes the celebrated Stirling approximation

[2]

(The symbol $\sim$ here means that the ratio of the two sides approaches 1 as $n$ becomes large. It is an asymptotic statement, not an exact equality.)

Gamma therefore does not merely fill in fractional factorials. It provides an analytic setting in which factorial-like growth can be studied continuously.

Gamma became one of mathematics' standard tools

There is no single modern subject called “the application of Gamma”. It appears in too many places for that. In probability and statistics, Gamma normalises the gamma distribution. In one common scale convention,

The denominator is what makes the total probability equal to one. In geometry, Gamma lets one formula describe the volume of a ball in any positive integer dimension:

(For $n=2$ this becomes the area $\pi R^2$. For $n=3$ it becomes the familiar sphere volume $4\pi R^3/3$. Gamma is what lets the same formula move cleanly through even and odd dimensions.)

In combinatorics and analysis, ratios such as

generalise rising products. In mathematical physics, differential equations and integral transforms, Gamma repeatedly appears when powers are integrated against exponential or Gaussian decay. It became what mathematicians call a special function — although “special” can be misleading. Functions such as Gamma are special partly because they are so useful that mathematics has given them names.

Gamma can evaluate integrals that otherwise look unrelated

Suppose we meet an integral of the form

A substitution turns it into

Gaussian-type integrals can be transformed in the same spirit. [2] That is one reason the function spreads so widely through mathematics. Once an integral can be transformed into Gamma's standard shape, a difficult-looking calculation becomes an evaluation of a known function.

And now Gamma meets zeta

This is where our story returns to the Riemann Hypothesis. There is an integral representation of the Riemann zeta function:

Equivalently,

[4] The two functions are now literally appearing in the same formula. This is not an artificial association invented for the Riemann Hypothesis. Gamma and zeta were already part of the same world of infinite sums, products, integrals and analytic continuation. Euler had worked on both.

Why Gamma appears in the completed zeta function

In What is the Riemann xi function?, we met

At first sight, the Gamma factor can look like an arbitrary correction attached to zeta. It is not. The combination

arises naturally when zeta is connected to Gaussian and theta-function integrals. Gamma is exactly the sort of function that appears when powers are integrated against exponential or Gaussian decay. In the completed zeta function, the Gamma factor belongs to the analytic machinery producing zeta's reflection symmetry. With the completion assembled, the more complicated functional equation of zeta becomes the beautifully simple symmetry

 FACTORIALS
     |
     | extend the recurrence
     v
 GAMMA FUNCTION
     |
     | integral + complex continuation
     v
 Gamma(s/2) * pi^(-s/2)
     |
     | joins zeta in the completion
     v
 RIEMANN XI FUNCTION
     |
     | clean reflection symmetry
     v
       xi(s) = xi(1-s)

This is one of the historical pleasures of the subject. Euler was not inventing a component for Riemann's later theory. Riemann would not be born for almost another century. Euler was solving a different problem. But the mathematical object he uncovered turned out to have exactly the analytic properties needed in a much later theory of prime numbers.

A discovery can become more important than the problem that produced it

The original question was easy to state: what should a factorial mean between the integers? Euler's answer did much more than fill the gaps. It created a function with a recurrence relation, an integral representation, product representations, reflection identities, analytic continuation and deep connections to other special functions. Later mathematicians discovered that Gamma helps organise probability distributions, high-dimensional geometry, asymptotic approximations, differential equations, integral transforms and complex analysis. And in Riemann's theory, it became one of the factors that reveals the hidden symmetry of zeta. This is a recurring pattern in mathematics. A problem asks for one thing. A good mathematical object answers it. A great mathematical object turns out to answer questions nobody had yet thought to ask.

The sequence to remember

The whole story can be compressed into a few steps. Factorials begin as a discrete recurrence:

Euler looks for a genuine function carrying that recurrence beyond the integers. In modern notation, his integral is

Integration by parts reveals

So

and

At the half-integers,

which gives

Analytic continuation carries Gamma through most of the complex plane. And eventually $\Gamma(s/2)$ becomes one of the ingredients in Riemann's completed zeta function.

The one sentence to remember

The gamma function is the canonical analytic extension of the factorial recurrence: Euler's integral obeys the same multiplication rule as factorials, extends it naturally beyond whole numbers, and becomes one of the fundamental special functions of mathematics. But perhaps the more memorable version is simply this:

The factorials were never confined to whole numbers after all.