### numbers.calculus

Simpson's method of approximating the integral of a function, f, on the interval [a,b].

Parameters f : Function The function to be evaluated. a : Number The left endpoint of the interval. b : Number The right endpoint of the interval. eps : Number An error bound. num : Number The approximation of the integral of f on [a,b]. This function does not raise any errors.

#### numbers.calculus.LanczosGamma(n)

Lanczos' approximation to the gamma function of a number, n.

Parameters n : Number A number. num : Number Gamma of n. This function does not raise any errors.

#### numbers.calculus.limit(f, x, approach)

Calculates the limit of a function, f, at a point x. The point can be approached from the left, right, or middle (a combination of the left and right).

Parameters f : Function The function to be evaluated. x : Number The point for which the limit will be calculated. approach : String A desired approach. left, right and middle are the possible approaches. num : Number The limit of f at x. An error is thrown if: an approach is not given

#### numbers.calculus.MonteCarlo(f, N, I)

The Monte-Carlo method for approximating the integral of a singlevariate or multivariate function, f, over a given interval(s). The number of intervals must match the number of variables of the function. The nth element of I is the interval for the nth variable of the function.

Parameters f : Function The function to be evaluated. N : Number The number of function evaluations. I : Array An array of arrays, where each inner array is an interval containing the endpoints. num : Number An approximation to the integral of f with N function evaluations. An error is thrown if: there are no intervals given (L.length == 0) N is not positive

#### numbers.calculus.pointDiff(f, x)

Calculates the point differential of a function f at a point x. Currently only supports one-dimensional functions.

Parameters f : Function The function to be evaluated. x : Number The point for which the point differential will be calculated. num : Number The point differential of f at x. This function does not raise any errors.

#### numbers.calculus.Riemann(f, a, b, n, sampler)

Calculates the Riemann sum for a one-variable function f on the interval [a,b] with n equally-spaced divisons. If sampler is given, that function will be used to calculate which value to sample on each subinterval; otherwise, the left endpoint will be used.

Parameters f : Function The function to be evaluated. a : Number The left endpoint of the interval. b : Number The right endpoint of the interval. n : Number The number of subintervals. sampler : Function, optional A function that determines what value is to be used for sampling on a subinterval. num : Number The Riemann sum of f on [a,b] with n divisions. This function does not raise any errors.

#### numbers.calculus.SimpsonDef(f, a, b)

Simpson's method of approximating the integral of a function, f, on the interval [a,b].

Parameters f : Function The function to be evaluated. a : Number The left endpoint of the interval. b : Number The right endpoint of the interval. num : Number The approximation of the integral of f on [a,b]. This function does not raise any errors.

#### numbers.calculus.SimpsonRecursive(f, a, b, whole, eps)

The helper function used for adaptive Simpson's method of approximating the integral of a function, f, on the interval [a,b].

Parameters f : Function The function to be evaluated. a : Number The left endpoint of the interval. b : Number The right endpoint of the interval. whole : Number The value of the integral of f on [a,b] (Simpson's approximation, in the caes of adaptiveSimpson). eps : Number An error bound. num : Number A recursive evaluation of the left and right sides. This function does not raise any errors.

#### numbers.calculus.StirlingGamma(n)

Striling's approximation to the gamma function of a number, n.

Parameters n : Number A number. num : Number Gamma of n. This function does not raise any errors.