Abstract
We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first N terms of the series. We show several examples of its application in calculating the inverses of some special functions.
Highlights
The existence of series expansions for inverses of analytic functions is a well-known result of complex analysis [17]
We present a simple, easy to implement method for computing the series expansion for the inverse of any function satisfying the conditions of Theorem 1.1, the method is especially powerful when h(x) has the form
We test our result with some known results and we apply the method for obtaining expansions for the inverse of the error function, the incomplete gamma function, the sine integral, and other special functions
Summary
The existence of series expansions for inverses of analytic functions is a well-known result of complex analysis [17]. The standard inverse function theorem, a proof of which can be found, for example, in [12], is as follows. We present a simple, easy to implement method for computing the series expansion for the inverse of any function satisfying the conditions of Theorem 1.1, the method is especially powerful when h(x) has the form. We test our result with some known results and we apply the method for obtaining expansions for the inverse of the error function, the incomplete gamma function, the sine integral, and other special functions
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