The solution of the problem of integration in finite terms
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Cites background from "The solution of the problem of inte..."
...In 1970, Risch [245] showed that this problem could be solved if one could decide if a given divisor on a given algebraic curve is of finite order....
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Cites background or methods from "The solution of the problem of inte..."
...For rational functions with only simple poles, this algorithm first appears in Johann Bernoulli (1703). For symbolic computation, this approach is inefficient since it involves polynomial factorization and computation with algebraic numbers, and most computer algebra systems implement the algorithms described in this chapter. In fact, on the pages just preceding Bernoulli’s article, Leibniz (1703) goes a step further....
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...For information about the highly active area of symbolic-numeric computations see the Special Issue of the Journal of Symbolic Computation (Watt & Stetter 1998) and Corless, Kaltofen & Watt (2003). 2.4. The first algorithm for division with remainder of polynomials appears in Nuñez (1567). He is, of course, limited by the concepts of his times to specific degrees, 3 and 1 in his case, and positive coefficients....
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...Knuth (1993) explains Johann Faulhaber’s (1631) methods for summation of powers, yielding much more beautiful expressions than the ones we give....
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...π had been used by Jones (1706) and by Christian Goldbach in 1742....
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...Knuth (1993) explains Johann Faulhaber’s (1631) methods for summation of powers, yielding much more beautiful expressions than the ones we give. A true renaissance man. 23.3. The definition of the gff, as well as Theorem 23.12 and Algorithm 23.13, are from Paule (1995). We have adopted the graphical representation of the shift structure from Pirastu (1992)....
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References
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"The solution of the problem of inte..." refers background in this paper
...(1), (2) and (3) are all the operations needed since we get the trigonometric and inverse trigonometric operations by adjoining V ( - l ) toK....
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...If in (2) or (3) 0t- is transcendental over 3D(0i, • • • , 0,_i) and the constant field of 3D(0i, • • • ,0;_i) is the same as that of 3D(0i, • • • , 0*), then 0» is a monomial over 3D(0i, • • • , 0t-x)....
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...If 3) is a differential subfield of $, then ^ (and any /G*?) is said to be elementary over 3D iff 3F = 3D(0i, • • • , 0n) where each 0t- satisfies a t least one of the following conditions: (1) 6i is algebraic over 3D(0i, • • • , 0,~i), (2) 9'i/di•=ƒ' for some/G3D(0x, • • • , 0i_i) (the exponential case), (3) f/f=0't for some ƒ G 3D (0i, • • • , 0*_i) (the logarithmic case)....
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