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F_x[cos(2pik_0x)](k) = int_(-infty)^inftye^(-2piikx)((e^(2piik_0x)+e^(-2piik_0x))/2)dx (1) = 1/2int_(-infty)^infty[e^(-2pii(k-k_0)x)+e^(-2pii(k+k_0)x)]dx (2) = ...

2^(30)=1024^3=1073741824 bytes. Although the term gigabyte is sometimes used to refer to 1024^3 bytes, such usage is deprecated in favor of the standard SI naming convention ...

Lagrange's continued fraction theorem, proved by Lagrange in 1770, states that any positive quadratic surd sqrt(a) has a regular continued fraction which is periodic after ...

Let each sphere in a sphere packing expand uniformly until it touches its neighbors on flat faces. Call the resulting polyhedron the local cell. Then the local density is ...

The illusion illustrated above in which the bounding rectangle and inner square both appear distorted.

The Pell constant is the infinite product P = 1-product_(k=0)^(infty)(1-1/(2^(2k+1))) (1) = 1-(1/2;1/4)_infty (2) = 0.58057755820489... (3) (OEIS A141848), where (a,q)_infty ...

An ideal I of a ring R is called principal if there is an element a of R such that I=aR={ar:r in R}. In other words, the ideal is generated by the element a. For example, the ...

The partial differential equation u_(tt)-u_(xx)=epsilon(u_t-u_t^3).

The function f(x,y)=(1-x)^2+100(y-x^2)^2 that is often used as a test problem for optimization algorithms (where a variation with 100 replaced by 105 is sometimes used; ...

An operator T which commutes with all shift operators E^a, so TE^a=E^aT for all real a in a field. Any two shift-invariant operators commute.