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Let z be defined as a function of w in terms of a parameter alpha by z=w+alphaphi(z). (1) Then Lagrange's inversion theorem, also called a Lagrange expansion, states that any ...

Let z=re^(itheta)=x+iy be a complex number, then inequality |(zexp(sqrt(1-z^2)))/(1+sqrt(1-z^2))|<=1 (1) holds in the lens-shaped region illustrated above. Written explicitly ...

The intersection of two lines L_1 and L_2 in two dimensions with, L_1 containing the points (x_1,y_1) and (x_2,y_2), and L_2 containing the points (x_3,y_3) and (x_4,y_4), is ...

In the fields of functional and harmonic analysis, the Littlewood-Paley decomposition is a particular way of decomposing the phase plane which takes a single function and ...

The Löwenheim-Skolem theorem is a fundamental result in model theory which states that if a countable theory has a model, then it has a countable model. Furthermore, it has a ...

Several prizes are awarded periodically for outstanding mathematical achievement. There is no Nobel Prize in mathematics, and the most prestigious mathematical award is known ...

The 2-cusped epicycloid is called a nephroid. The name nephroid means "kidney shaped" and was first used for the two-cusped epicycloid by Proctor in 1878 (MacTutor Archive). ...

Orthogonal circles are orthogonal curves, i.e., they cut one another at right angles. By the Pythagorean theorem, two circles of radii r_1 and r_2 whose centers are a ...

The pentagram, also called the five-point star, pentacle, pentalpha, or pentangle, is the star polygon {5/2}. It is a pagan religious symbol that is one of the oldest symbols ...

pi may be computed using a number of iterative algorithms. The best known such algorithms are the Archimedes algorithm, which was derived by Pfaff in 1800, and the ...