Abstract
We call algebraic functions those that we form by only employing the first operations of the algebra, namely, addition, subtraction, multiplication, division, and the raising of variables to fixed powers. The algebraic functions of one variable are rational when they contain only integer powers of this variable, that is to say, when they are reduced to entire functions or to rational fractions. They are irrational otherwise.
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Notes
- 1.
The second expression in (5) from the 1899 edition reads,
$$\begin{aligned} \varUpsilon \left[ x, \, \sqrt{(a_1x+b_1)^2+(a_2x+b_0)(a_0x+b_2)} \right] \, dx. \end{aligned}$$Additionally, in the original 1823 edition, this same expression reads,
$$\begin{aligned} \varUpsilon \left[ x, \, \sqrt{(a_1x+b_1)^2+(a_0x+b_0)(a_0x+b_2)} \right] \, dx. \end{aligned}$$Both of these misprints have been corrected here.
- 2.
Both of Cauchy’s texts reference equation number (17), a clear misprint that has been corrected here.
- 3.
Cauchy references equation (16), another simple error which has been corrected.
- 4.
Parentheses have been added in the third expression.
- 5.
An incredible remark indeed.
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Cates, D.M. (2019). ON INDEFINITE INTEGRALS WHICH CONTAIN ALGEBRAIC FUNCTIONS.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_28
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DOI: https://doi.org/10.1007/978-3-030-11036-9_28
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Online ISBN: 978-3-030-11036-9
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