Експоненцијална функција — разлика између измена
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Ред 36:
The [[graph of a function|graph]] of <math>y=e^x</math> is upward-sloping, and increases faster as {{mvar|x}} increases. The graph always lies above the {{mvar|x}}-axis but can be arbitrarily close to it for negative {{mvar|x}}; thus, the {{mvar|x}}-axis is a horizontal [[asymptote]]. The [[slope]] of the [[tangent]] to the graph at each point is equal to its {{mvar|y}}-coordinate at that point, as implied by its derivative function (''see above''). Its [[inverse function]] is the [[natural logarithm]], denoted <math>\log,</math><ref>In pure mathematics, the notation {{math|log ''x''}} generally refers to the natural logarithm of {{mvar|x}} or a logarithm in general if the base is immaterial.</ref> <math>\ln,</math><ref>The notation {{math|ln ''x''}} is the ISO standard and is prevalent in the natural sciences and secondary education (US). However, some mathematicians (e.g., [[Paul Halmos]]) have criticized this notation and prefer to use {{math|log ''x''}} for the natural logarithm of {{mvar|x}}.</ref> or <math>\log_e;</math> because of this, some old texts<ref>{{cite book |quote=Inverse Use of a Table of Logarithms; that is, given a logarithm, to find the number corresponding to it, (called its antilogarithm) ... |page=12 |last=Converse |last2=Durrell |title=Plane and spherical trigonometry |publisher=C. E. Merrill Co. |year=1911 |url=https://books.google.com/books?id=xTIAAAAAYAAJ&pg=PA12 }}</ref> refer to the exponential function as the '''antilogarithm'''.
== Формална дефиниција ==▼
{{main|Characterizations of the exponential function}}
[[Image:Exp series.gif|right|thumb|The exponential function (in blue), and the sum of the first {{math|''n'' + 1}} terms of its power series (in red).]]
Експоненцијална функција -{e}-<sup>''-{x}-''</sup> може се дефинисати на доста еквивалентних начина, преко бесконачних редова. Одређеније, може се дефинисати преко [[Степени ред|степених редова]]:<ref name="rudin" />▼
: <math>e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots</math>▼
ince the [[radius of convergence]] of this power series is infinite, this definition is, in fact, applicable to all complex numbers <math>z \in \mathbb{C}</math> (see [[Exponential function#Complex plane|below]] for the extension of <math>\exp(x)</math> to the complex plane). The constant {{mvar|e}} can then be defined as <math display="inline">e=\exp(1)=\sum_{k=0}^\infty(1/k!).</math>
The term-by-term differentiation of this power series reveals that <math>(d/dx) (\exp x) = \exp x</math> for all real {{mvar|x}}, leading to another common characterization of <math>\exp(x)</math> as the unique solution of the [[differential equation]]
:<math>y'(x)=y(x),</math>
satisfying the initial condition <math>y(0)=1.</math>
Based on this characterization, the [[chain rule]] shows that its inverse function, the [[natural logarithm]], satisfies <math>(d/dy) (\log_e y) = 1/y</math> for <math>y>0,</math> or <math display="inline">\log_e y=\int_1^y \frac{1}{t}\,dt.</math> This relationship leads to a less common definition of the real exponential function <math>\exp(x)</math> as the solution <math>y</math> to the equation
: <math>x = \int_1^y \frac{1}{t} \, dt.</math>
By way of the [[binomial theorem]] and the power series definition, the exponential function can also be defined as the following limit:<ref name="autogenerated156">[[Eli Maor]], ''e: the Story of a Number'', p.156.</ref>
У овим дефиницијама, <math>n!</math> означава [[факторијел]] броја ''-{n}-'', а ''-{x}-'' је или произвољан реалан број, комплексан број, елемент Банахове алгебре (на пример, квадратна матрица).▼
== Својства ==
Линија 93 ⟶ 117:
: <math>{d \over dx} e^{f(x)} = f'(x)e^{f(x)}</math>
▲== Формална дефиниција ==
▲Експоненцијална функција -{e}-<sup>''-{x}-''</sup> може се дефинисати на доста еквивалентних начина, преко бесконачних редова. Одређеније, може се дефинисати преко степених редова:
▲: <math>e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots</math>
▲: <math>e^x = \lim_{n \to \infty} \left(1 + {x \over n} \right)^n.</math>
▲У овим дефиницијама, <math>n!</math> означава [[факторијел]] броја ''-{n}-'', а ''-{x}-'' је или произвољан реалан број, комплексан број, елемент Банахове алгебре (на пример, квадратна матрица).
== Нумеричка вредност ==
Линија 164 ⟶ 176:
<ref name="rudin">{{cite book|url=https://archive.org/details/RudinW.RealAndComplexAnalysis3e1987|title=Real and complex analysis|date=1987|publisher=McGraw-Hill|isbn=978-0-07-054234-1|edition=3rd|location=New York|page=1|quote=|via=|last1=Rudin|first1=Walter}}</ref>
}}
== Литература ==
{{Refbegin}}
* [[Walter Rudin]], ''Principles of Mathematical Analysis'', 3rd edition (McGraw–Hill, 1976), chapter 8.
* [[Edwin Hewitt]] and Karl Stromberg, ''Real and Abstract Analysis'' (Springer, 1965).
{{Refend}}
== Спољашње везе ==
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