Why certain integrals are ``impossible'.

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IntroductionElementary functions and fieldsLiouville s TheoremAn exampleWhy certain integrals are impossible .Pete GoetzDepartment of MathematicsSonoma State UniversityMarch 11, 2009IntroductionElementary functions and fieldsLiouville s TheoremAn fields and s functions and fieldsLiouville s TheoremAn exampleProbabilityCentral Limit Theorem (x)=1 2 x e u2/2duFor probability applications, we need ( ) = is not proved by finding a formula for (x) (by findingan explicit antiderivative ofe u2/2) and taking the limit asx .IntroductionElementary functions and fieldsLiouville s TheoremAn exampleNumber TheoryPrime Number Theorem (x) = #{n x|nis prime}Li(x)= x21ln(t)dt (x) Li(x) asx This is not proved by finding an explicit antiderivative of1ln(t).Ifu= ln(t), then 1ln(t)dt= functions and fieldsLiouville s TheoremAn exampleElementary formulasThe indefinite integrals e u2duand euududo not haveelementary does one prove such claims?

Introduction Elementary Functions and fields Liouville’s Theorem An example Elementary fields A field K is an elementary field if K = C(x,f1,...,fn) and each fj is an exponential or logarithm of an element of

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