Special Functions


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If u is a column vector and m is a row vector, the results are matrices with length u rows and length m columns. Otherwise, u and m must conform in size and the results will be the same size as the inputs. The value of u may be complex. If requested, err contains the following status information and is the same size as the result. Compute complete elliptic integrals of the first K m and second E m kind. The optional input tol controls the stopping tolerance of the algorithm and defaults to eps class m.

The tolerance can be increased to compute a faster, less accurate approximation.

Special Functions

Reference: Milton Abramowitz and Irene A. See also: erfc , erfcx , erfi , dawson , erfinv , erfcinv. See also: erfcinv , erfcx , erfi , dawson , erf , erfinv. See also: erfc , erf , erfi , dawson , erfinv , erfcinv. See also: erfc , erf , erfcx , dawson , erfinv , erfcinv. See also: erf , erfc , erfcx , erfi , dawson , erfcinv. See also: erfc , erf , erfcx , erfi , dawson , erfinv. See also: cosint , sinint , exp.

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Programming Note: The gamma function can grow quite large even for small input values. In many cases it may be preferable to use the natural logarithm of the gamma function gammaln in calculations to minimize loss of precision. See also: gammainc , gammaln , factorial. The standard notation is P a,x , e.

If a is scalar, then gammainc x , a is returned for each element of x and vice versa. If neither x nor a is scalar then the sizes of x and a must agree, and gammainc is applied element-by-element. The elements of a must be non-negative. By default, tail is "lower" and the incomplete gamma function integrated from 0 to x is computed.

If tail is "upper" then the complementary function integrated from x to infinity is calculated. If tail is "scaledupper" , then the upper incomplete gamma function is multiplied by the same quantity. Abramowitz and I. Stegun, Handbook of mathematical functions , Dover publications, Inc.

Math Software, pp.


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Flannery, Numerical Recipes in Fortran 77 , ch. See also: gamma , gammaincinv , gammaln. If a is scalar then gammaincinv y , a is returned for each element of y and vice versa. If neither y nor a is scalar then the sizes of y and a must agree, and gammaincinv is applied element-by-element. The variable y must be in the interval [0,1] while a must be real and positive.

By default, tail is "lower" and the inverse of the incomplete gamma function integrated from 0 to x is computed. If tail is "upper" , then the complementary function integrated from x to infinity is inverted. Gil, J. Segura, and N.

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Temme, Efficient and accurate algorithms for the computation and inversion of the incomplete gamma function ratios , SIAM J. Computing, pp. A—A, Vol 34, See also: gammainc , gamma , gammaln. The optional argument normalization may be one of "unnorm" , "sch" , or "norm".

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The default if no normalization is given is "unnorm". The return value has one dimension more than x.


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  • When the optional argument normalization is "sch" , compute the Schmidt semi-normalized associated Legendre function. The Schmidt semi-normalized associated Legendre function is related to the unnormalized Legendre functions by the following:. When the optional argument normalization is "norm" , compute the fully normalized associated Legendre function.

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    Special Functions

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