Euler product formula

The Euler product formula can be used to calculate the asymptotic probability that s randomly selected integers are set-wise coprime. Intuitively, the probability that any single number is divisible by a prime (or any integer), p is 1/p.

Hence the probability that s numbers are all divisible by this prime is 1/p^s, and the probability that at least one of them is not is 1 − 1/p^s. Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisors n and m if and only if it is divisible by nm, an event which occurs with probability 1/(nm).)

Thus the asymptotic probability that s numbers are coprime is given by a product over all primes,

Gamma, the Euler–Mascheroni constant.

Gamma, the Euler–Mascheroni constant. 

The number γ has not been proved algebraic or transcendental. In fact, it is not even known whether γ is irrational. Continued fraction analysis reveals that if γ is rational, its denominator must be greater than 10242080. The ubiquity of γ revealed by the large number of equations makes the irrationality of γ a major open question in mathematics.

Integral of rational function

From http://en.wikipedia.org/wiki/Catenary:

The word catenary is derived from the Latin word catena, which means "chain". The English word catenary is usually attributed to Thomas Jefferson, who wrote in a letter to Thomas Paine on the construction of an arch for a bridge:

I have lately received from Italy a treatise on the equilibrium of arches, by the Abbé Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect equilibrium.^[5]

It is often said that Galileo thought the curve of a hanging chain was parabolic. In his Two New Sciences (1638), Galileo says that a hanging cord is an approximate parabola, and he correctly observes that this approximation improves as the curvature gets smaller and is almost exact when the elevation is less than 45°. That the curve followed by a chain is not a parabola was proven by Joachim Jungius (1587–1657); this result was published posthumously in 1669.

de Moivre's formula

de Moivre's formula 

From http://en.wikipedia.org/wiki/De_Moivre's_formula:

By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for cos (nx) and sin (nx) in terms of cos x and sin x. Furthermore, one can use a generalization of this formula to find explicit expressions for the nth roots of unity, that is, complex numbers z such that zn = 1.

Legendre's differential equation

They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates.

The Legendre differential equation may be solved using the standard power series method. The equation has regular singular points at x = ±1 so, in general, a series solution about the origin will only converge for |x| < 1. When n is an integer, the solution Pn(x) that is regular at x = 1 is also regular at x = −1, and the series for this solution terminates (i.e. is a polynomial).

The Sophomore's dream

In mathematics, sophomore's dream is a name occasionally used for the above given identity discovered in 1697 by Johann Bernoulli.

Golden ratio countinuos fraction expansion

From http://en.wikipedia.org:

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.

Jean-Baptiste Mondino (born Aubervilliers, France in 1949) is a French fashion photographer and music video director. He has directed music videos for Madonna, David Bowie, Sting, Björk, China Moses (Dee Dee Bridgewater's daughter) and Les Rita Mitsouko.

Dirac delta function

From a purely mathematical viewpoint, the Dirac delta is not strictly a function, because any real function that is equal to zero everywhere but a single point must have total integral zero.

While for many purposes the Dirac delta can be manipulated as a function, formally it can be defined as a distribution that is also a measure. In many applications, the Dirac delta is regarded as a kind of limit (a weak limit) of a sequence of functions having a tall spike at the origin.

Arcsin integral

Bring back the good old high school memories, One of my favorite hyperbolic integrals.

Sine function Taylor serie expansion

Euler's identity (tau notation)

Euler formula in Tau notation

One of the most astounding formula ever discovered, setting the foundation of complex mathematical analysis, here presented in tau (turns) notation and displaying, in my opinion, the five most important numbers in the history of mathematics.