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Let g(x)=(1-x^2)(1-k^2x^2). Then int_0^a(dx)/(sqrt(g(x)))+int_0^b(dx)/(sqrt(g(x)))=int_0^c(dx)/(sqrt(g(x))), where c=(bsqrt(g(a))+asqrt(g(b)))/(sqrt(1-k^2a^2b^2)).

Let f(z) be an analytic function in |z-a|<R. Then f(z)=1/(2pi)int_0^(2pi)f(z+re^(itheta))dtheta for 0<r<R.

U_n(f)=int_a^bf(x)K_n(x)dx, where {K_n(x)} is a sequence of continuous functions.

Always increasing; never remaining constant or decreasing. Also called strictly increasing.

A sequence {a_n} such that either (1) a_(i+1)>=a_i for every i>=1, or (2) a_(i+1)<=a_i for every i>=1.

An integral which has neither limit infinite and from which the integrand does not approach infinity at any point in the range of integration.

The value at a stationary point.

If a continuous function defined on an interval is sometimes positive and sometimes negative, it must be 0 at some point. Bolzano (1817) proved the theorem (which effectively ...

Suppose f(x) is continuous at a stationary point x_0. 1. If f^'(x)>0 on an open interval extending left from x_0 and f^'(x)<0 on an open interval extending right from x_0, ...

A function is said to be piecewise constant if it is locally constant in connected regions separated by a possibly infinite number of lower-dimensional boundaries. The ...