# What is the Gamma Function?

A lot of important functions in applied sciences are defined using improper integrals, and perhaps the most famous of these is the Gamma Function. Throughout my course I have heard mention of this function, and always wondered “but what exactly is the Gamma Function?” I finally decided to find out.

Mathematicians wanted to find a formula that generalised the factorial expression for the natural numbers. This lead to the discovery of following very well-known formula:

Say we replace the n but the x to create the function:

What is the domain of this function? The only possible ‘bad points‘ are 0 and ∞:

• t ≈ 0: e^-t ≈ 1 and hence the expression inside the integral is approximately t^x, meaning we have convergence at around 0, only if x>-1.
• t –> ∞: No matter the value of x, the integral is convergent.

From this we can see that the domain of f(c)  is (-1,∞).

To obtain (0,∞) as the domain, we need to shift the function one unit to the right. This defines the Gamma Function.

The above discussion explains why the Gamma Function has the x-1 power.

### Some Properties:

• Γ(n) = (n-1)! where n is a natural number
• Γ(x+1) = xΓ(x) for x > 0. This follows easily from the definition of Γ(x) by performing integration by parts.
• Weierstrass’s definition is also valid for all complex numbers (except non-positive integers):where is the Euler-Mascheroni Constant.
• Taking the logarithm of the above expression (we can do this as Γ(x) > 0 for x > 0), we get that:

• Differentiating this, we observe that:In fact, Γ(x) can be differentiated infinitely often.
• Γ(x) is connected with sin(x): For x > 0

M x

# MATHS BITE: Even or Odd?

What are even and odd functions? These two terms get used very frequently in order to simplify problems such as integration or when finding Fourier coefficients, for example.

### Even Function

An even function is such that for all x in the domain

f(x) = f(-x)

An even function is symmetric with respect to the y-axis.

Even Function | mathsisfun

Examples:

• Any polynomial p, where n is even for all xn. For example f(x) = x2+2x4 – 6. Note that (x+1)2 is not even.
• f(x) = cos(x)

### Odd Function

An odd function is such that for all x in the domain

f(x) = -f(-x)

An odd function is symmetric with respect to the origin.

Odd Function | mathsisfun

Examples:

• x, x3, x5,… etc. Note that unlike even functions, an expression such as x3 + 1 is not odd.
• f(x) = sin(x)

### Odd and/or Even?

A function can be neither odd nor even, for example x− x + 1:

Neither odd nor even | mathsisfun

Additionally, the only function that is even and odd is f(x) = 0.

### Properties:

• Adding two even (odd) functions will give an even (odd) function.
• Adding an even and an odd function will give a function that is neither odd nor even.
• Both the product of two even functions and the product of two odd functions is even.
• The product of an even function and an odd function is odd.

# Noether’s Theorem

Today I thought I’d write a blog post about an interesting theorem I learnt whilst studying my Variational Principles module – Noether’s Theorem.

To understand Noether’s Theorem, we must first understand what is meant by a symmetry of a functional.

Given

suppose we change the variables by the transformation t –> t*(t) and x –> x*(t) to obtain a new independent variable and a new function. This gives

where α* = t*(α) and β* = t*(β).

If F*[x*] = F[x] for all x, α and β, then this transformation * is called a symmetry.

What is a continuous symmetry?

Intuitively, a continuous symmetry is a symmetry that we can do a bit ofFor example, a rotation is a continuous symmetry, but a reflection is not.

### Noether’s Theorem

Noether’s Theorem – proven by mathematician Emmy Noether in 1915 and published in 1918 – states that every continuous symmetry of F[x] the solutions (i.e. the stationary points of F[x]) will have a corresponding conserved quantity.

### Why?

Consider symmetries that involve only the x variable. Then, up to first order, the symmetry can be written as:

t –> t, x(t) –> x(t) + εh(t)

where h(t) represents the symmetry transformation. As the transformation is a symmetry, we can pick ε to be any small constant number and F[x] does not change, i.e. δF = 0. Also, since x(t) is a stationary point of F[x], we know that if ε is any non-constant, but vanishes at the end-points, then we have δF = 0 again. Combining these two pieces of information, we can show that there is a conserved quantity in the system.

For now, do not make any assumptions about ε. Under the transformation, the change in F[x] is given by

Firstly, consider the case where ε is constant. Then the second integral vanishes and we obtain

So we know that

Now, consider a variable ε that is not constant, but vanishes at the endpoints. Then, as is a solution, we must have that δF = 0. Therefore,

If we integrate the above expression by parts, we get that

Hence the conserved quantity is:

Not all symmetries involve just the x variable, for example we may have a time translation, but we can encode this as a transformation of the x variable only.

M x

# MATHS BITE: Apéry’s Constant

Apéry’s constant is defined as the number

where ζ is the Riemann Zeta Function.

This constant is named after the French mathematician Roger Apéry who proved that it was irrational in 1978. However it is still unknown whether or not it is transcendental.