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Step: This is an exponential decay function where is a constant greater than or equal to 2. As n → ∞ {\displaystyle n\to \infty } , a − n → 0 {\displaystyle a^{-n}\to 0} very quickly, making it negligible.
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
In mathematics, the Laurent series of a complex function is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.
The converse, though, does not necessarily hold: for example, taking f as =, where V is a Vitali set, it is clear that f is not measurable, but its absolute value is, being a constant function. The positive part and negative part of a function are used to define the Lebesgue integral for a real-valued function.
Bernoulli's inequality can be proved for case 2, in which is a non-negative integer and , using mathematical induction in the following form: we prove the inequality for r ∈ { 0 , 1 } {\displaystyle r\in \{0,1\}} ,
For negative real radicands, and odd exponents, the principal n th root is not real, although the usual n th root is real. Analytic continuation shows that the principal n th root is the unique complex differentiable function that extends the usual n th root to the complex plane without the nonpositive real numbers.
However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g., tan −1 = arctan ≠ 1/tan. In some cases, when, for a given function f, the equation g ∘ g = f has a unique solution g, that function can be defined as the functional square root of f, then written as g = f 1/2.
Exponential functions occur very often in solutions of differential equations. The exponential functions can be defined as solutions of differential equations. Indeed, the exponential function is a solution of the simplest possible differential equation, namely ′ = .
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