The Impossibility of Integrating x^xDate: 08/07/98 at 09:47:58 From: Nick Black Subject: Symbolic integration of x^x...impossible? Hello. I am a 17 year-old at Walton High School, and last year I took AP Calculus BC (covering differential, integral, and series-based calculus). I found it an enjoyable and intriguing class, but there was one problem I could not solve. Misreading a question as asking for the symbolic integral of x^x, as opposed to the differential, I spent most of the day trying to generate a true integral. The differential was easy, of course. It equals x^x(ln(x)+1). In any case, I have tried converting the equation to a Taylor polynomial and creating a factorial-based polyterminal equation, converting to a summation, and generating a Rienmann sum, but ater a while, without seeing any discernible pattern, I get bogged down in the math (the 8th-order differential of x^x is an ugly beast). I hope this is within the realm of your service. Thank you. Date: 08/09/98 at 08:34:48 From: Doctor Jerry Subject: Re: Symbolic integration of x^x...impossible? Hi Nick, There is an algorithm attributed to a contemporary mathematician named Risch that can decide whether the anti-derivative of a continuous function f can be expressed as a finite combination of elementary functions. The elementary functions include polynomials, the trig functions, the inverse trig functions, the exponential function and its inverse, etc. There are many functions - called special functions - that fail to have an anti-derivative expressible as a finite combination of elementary functions. The so-called elliptic functions, the error function, and the gamma function are a few examples. The error function, which is extremely useful in both physics and statistics, is defined as: erf(x) = (2/sqrt(pi)) integral from 0 to x of e^(-t^2)dt There would not exist extensive tables of the error function if the anti-derivative of e^(-t^2) were expressible as a finite combination of elementary functions. The anti-derivative of x^x is not expressible as a finite combination of elementary functions. I'm not sufficiently familiar with the Risch algorithm to even hint at a proof of it. - Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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