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For R[mu+nu]>1, int_(-pi/2)^(pi/2)cos^(mu+nu-2)thetae^(itheta(mu-nu+2xi))dtheta=(piGamma(mu+nu-1))/(2^(mu+nu-2)Gamma(mu+xi)Gamma(nu-xi)), where Gamma(z) is the gamma function.

The exponential sum function e_n(x), sometimes also denoted exp_n(x), is defined by e_n(x) = sum_(k=0)^(n)(x^k)/(k!) (1) = (e^xGamma(n+1,x))/(Gamma(n+1)), (2) where ...

The functional derivative is a generalization of the usual derivative that arises in the calculus of variations. In a functional derivative, instead of differentiating a ...

A generalization of the product rule for expressing arbitrary-order derivatives of products of functions, where (n; k) is a binomial coefficient. This can also be written ...

A function f is said to have a lower bound c if c<=f(x) for all x in its domain. The greatest lower bound is called the infimum.

Let R(x) be the revenue for a production x, C(x) the cost, and P(x) the profit. Then P(x)=R(x)-C(x), and the marginal profit for the x_0th unit is defined by ...

P_n(cosalpha)=(sqrt(2))/piint_0^alpha(cos[(n+1/2)phi])/(sqrt(cosphi-cosalpha))dphi, where P_n(x) is a Legendre polynomial.

A problem in the calculus of variations. Let a vessel traveling at constant speed c navigate on a body of water having surface velocity u = u(x,y) (1) v = v(x,y). (2) The ...

If del xF=0 (i.e., F(x) is an irrotational field) in a simply connected neighborhood U(x) of a point x, then in this neighborhood, F is the gradient of a scalar field phi(x), ...

The derivative of the power x^n is given by d/(dx)(x^n)=nx^(n-1).