function

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:

displaymath227

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

displaymath233

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.

displaymath261

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

GammaFunction

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):{\displaystyle \Gamma (z)={\frac {e^{-\gamma z}}{z}}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)^{-1}e^{z/n}}where{\displaystyle \gamma \approx 0.577216} 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:

displaymath351

  • Differentiating this, we observe that:displaymath355In fact, Γ(x) can be differentiated infinitely often.
  • Γ(x) is connected with sin(x): For x > 0 displaymath451

Click here for more properties and applications.

M x

Advertisements

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.

Screen Shot 2017-10-11 at 5.20.23 PM

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)

Screen Shot 2017-10-11 at 5.24.53 PM.png

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.

Screen Shot 2017-10-11 at 5.20.18 PM

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)

Screen Shot 2017-10-11 at 5.30.14 PM.png

Odd and/or Even?

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

Screen Shot 2017-10-11 at 5.31.18 PM.png

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

Screen Shot 2017-08-08 at 11.45.48 AM.png

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 givesScreen Shot 2017-08-08 at 11.45.51 AM.png

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.jpg

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

Screen Shot 2017-08-08 at 11.57.37 AM.png

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

Screen Shot 2017-08-08 at 11.58.21 AM.png

Screen Shot 2017-08-08 at 11.58.38 AM.png

So we know that

Screen Shot 2017-08-08 at 11.58.41 AM.png

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,

Screen Shot 2017-08-08 at 12.00.00 PM.png

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

Screen Shot 2017-08-08 at 12.00.03 PM.png

Hence the conserved quantity is:

Screen Shot 2017-08-08 at 12.01.56 PM.png

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

{\displaystyle {\begin{aligned}\zeta (3)&=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}\\&=\lim _{n\to \infty }\left({\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+\cdots +{\frac {1}{n^{3}}}\right)\end{aligned}}}

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.

History

The Basel Problem asked about the convergence of the following sum:
Screen Shot 2017-06-10 at 2.16.05 PM.png

In the 18th century, Leonhard Euler proved that in fact it did – to π^2/6. However, the limit of the following sum remained unknown:Screen Shot 2017-06-10 at 2.19.28 PM.png

Although mathematicians made some progress, including Euler who calculated the first 16 decimal digits of the sum, it was not known whether the number was rational or irrational, until Apéry.

Furthermore, it is currently not known specifically whether any other particular ζ(n), for n odd, is irrational. “The best we’ve got is from Wadim Zudilin, in 2001, who showed that at least one of ζ(5), ζ(7), ζ(9), ζ(11) must be irrational, and Tanguy Rivoal, in 2000, who showed that infinitely many of the ζ(2k+1) must be irrational.”

M x