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Then by substitution of this condition into the equation of the hyperbola, the desired conclusion will be proven. Example. This method can be applied to problem #6 at IMO 1988: Let a and b be positive integers such that ab + 1 divides a 2 + b 2. Prove that a 2 + b 2 / ab + 1 is a perfect square. Let a 2 + b 2 / ab + 1 = q and ...
Suppose further that a 1 /a 2 and a 0 /a 2 are analytic functions. The power series method calls for the construction of a power series solution = =. If a 2 is zero for some z, then the Frobenius method, a variation on this method, is suited to deal with so called "singular points". The method works analogously for higher order equations as ...
A method similar to Vieta's formula can be found in the work of the 12th century Arabic mathematician Sharaf al-Din al-Tusi. It is plausible that the algebraic advancements made by Arabic mathematicians such as al-Khayyam, al-Tusi, and al-Kashi influenced 16th-century algebraists, with Vieta being the most prominent among them. [2] [3]
In the context of arithmetic, a proof of the formula = is a computation reducing the two terms to the same numeral. Kolmogorov followed the same lines but phrased his interpretation in terms of problems and solutions. To assert a formula is to claim to know a solution to the problem represented by that formula.
The solutions in terms of the original variable are obtained by substituting x 3 back in for u, which gives x 3 = 1 and x 3 = 8. {\displaystyle x^{3}=1\quad {\text{and}}\quad x^{3}=8.} Then, assuming that one is interested only in real solutions, the solutions of the original equation are
Some authors allow any real , [1] [2] whereas others require that not be 0 or 1. [ 3 ] [ 4 ] The equation was first discussed in a work of 1695 by Jacob Bernoulli , after whom it is named. The earliest solution, however, was offered by Gottfried Leibniz , who published his result in the same year and whose method is the one still used today.
Bracket series notations are notations that substitute for common power series notations (Table 1). [19] Replacing power series notations with bracket series notations transforms the power series to a bracket series. A bracket series facilitates identifying the formula parameters needed for integration.
Hume-Rothery rules, named after William Hume-Rothery, are a set of basic rules that describe the conditions under which an element could dissolve in a metal, forming a solid solution. There are two sets of rules; one refers to substitutional solid solutions, and the other refers to interstitial solid solutions.