The file ExponentialIntegrals.mth defines functions for approximating exponential, logarithmic, sine, and cosine integrals.  The definitions, nomenclature, and organization in this section follow Abramowitz and Stegun's Handbook of Mathematical Functions, Dover Publications, Inc. [1965].  The function definitions in the file are automatically loaded when any of its functions are first used.


Many of the functions in this file are defined in terms of definite integrals for which you must use the Simplify > Approximate command1LCSBCN to determine an approximate numerical result for numerical values of the arguments.  In such cases the accuracy is usually close to the current working precision if you receive no  Dubious accuracy  warning message.  As a check you can temporarily increase the precision by five digits and compare the values.


In other cases these definitions are truncated series for which you must choose a particular number of terms.  Asymptotic series generally diverge but have an error less than the first omitted term, so you can start with one term and keep increasing the number of terms while the change is diminishing in magnitude.  In contrast, for a convergent series you can keep increasing the number of terms until there is no change in the displayed approximation.


In some cases you can determine other truncated series by expanding an integrand into an integrable truncated series, then integrating symbolically.


This section lists classic integral and series definitions for ease of identification.  However, some of the integrals or series are transformed in the files into mathematically equivalent ones that are faster and more accurate to approximate.


EI(x, m) approximates to m terms of a series for the exponential integral Ei(x) defined for x>0 as the Cauchy principal value of the integral of exp(-t)/t from t=-x to infinity.


LI(x, m) approximates to m terms of a series for the logarithmic integral li(x) defined for x>1 as the Cauchy principal value of the integral of 1/LN(t) from t=0 to x.


EN(n, z) approximates to the nth exponential integral En(z) defined for the real part of z>0 and n a nonnegative integer as the integral of exp(-z·t)/t^n from t=1 to infinity.


EN_ASYMP(n, z, m) approximates to m+1 terms of an asymptotic series that approximates the above nth exponential integral for large magnitude z.


EI1(z, m) approximates to m terms of a series for the 1st exponential integral E1(z) with |phase z| < π.


SI(z) approximates to the sine integral Si(z) defined as the integral of sin(t)/t from t=0 to z.


CI(z) approximates to the cosine integral Ci(z) with |phase z| < π defined as the integral of (cos(t)-1)/t from t=0 to z plus Euler's gamma constant plus LN(z).


Other Utility File LibraryG5BS2R 

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