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If the discriminant is zero the fraction converges to the single root of multiplicity two. If the discriminant is positive the equation has two real roots, and the continued fraction converges to the larger (in absolute value) of these. The rate of convergence depends on the absolute value of the ratio between the two roots: the farther that ...
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. [1]
When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication and placed as a superscript to the right of their base. [2] Thus 3 + 5 2 = 28 and 3 × 5 2 = 75. These conventions exist to avoid notational ambiguity while allowing notation to remain brief. [4]
In the first of these equations the ratio tends toward A n / B n as z tends toward zero. In the second, the ratio tends toward A n / B n as z tends to infinity. This leads us to our first geometric interpretation. If the continued fraction converges, the successive convergents A n / B n are eventually arbitrarily close ...
Examples include approaches to solving the heat equation, Schrödinger equation, wave equation, and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative which, together with the fractional integral , is one of the ...
The fractional Schrödinger equation, a fundamental equation of fractional quantum mechanics, has the following form: [69] [70] (,) = (,) + (,) (,). where the solution of the equation is the wavefunction ψ ( r , t ) – the quantum mechanical probability amplitude for the particle to have a given position vector r at any given time t , and ħ ...