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X subset= R^n is semianalytic if, for all x in R^n, there is an open neighborhood U of x such that X intersection U is a finite Boolean combination of sets {x^_ in ...

A conjecture that, as proved by Parshin (1968), implies the Mordell conjecture.

A spheroidal harmonic is a special case of an ellipsoidal harmonic that satisfies the differential equation d/(dx)[(1-x^2)(dS)/(dx)]+(lambda-c^2x^2-(m^2)/(1-x^2))S=0 on the ...

A square root of x is a number r such that r^2=x. When written in the form x^(1/2) or especially sqrt(x), the square root of x may also be called the radical or surd. The ...

If at least one of d, e, or f has the form q^(-N) for some nonnegative integer N (in which case both sums terminate after N+1 terms), then ...

Algebra

The following integral transform relationship, known as the Abel transform, exists between two functions f(x) and g(t) for 0<alpha<1, f(x) = int_0^x(g(t)dt)/((x-t)^alpha) (1) ...

Some authors define a general Airy differential equation as y^('')+/-k^2xy=0. (1) This equation can be solved by series solution using the expansions y = ...

A composite number n is a positive integer n>1 which is not prime (i.e., which has factors other than 1 and itself). The first few composite numbers (sometimes called ...

Denote the nth derivative D^n and the n-fold integral D^(-n). Then D^(-1)f(t)=int_0^tf(xi)dxi. (1) Now, if the equation D^(-n)f(t)=1/((n-1)!)int_0^t(t-xi)^(n-1)f(xi)dxi (2) ...