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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Inverse tangent integral ： ウィキペディア英語版 Polylogarithm In mathematics, the polylogarithm (also known as , for Alfred Jonquière) is a special function Li''s''(''z'') of order ''s'' and argument ''z''. Only for special values of ''s'' does the polylogarithm reduce to an elementary function such as the natural logarithm or rational functions. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams. The polylogarithm function is equivalent to the Hurwitz zeta function — either function can be expressed in terms of the other — and both functions are special cases of the Lerch transcendent. Polylogarithms should not be confused with polylogarithmic functions nor with the offset logarithmic integral which has a similar notation. The polylogarithm function is defined by the infinite sum, or power series: :$$ \operatorname_s(z) = \sum_^\infty = z + + + \cdots \,. This definition is valid for arbitrary complex order ''s'' and for all complex arguments ''z'' with |''z''| < 1; it can be extended to |''z''| ≥ 1 by the process of analytic continuation. The special case ''s'' = 1 involves the ordinary natural logarithm, Li1(''z'') = −ln(1−''z''), while the special cases ''s'' = 2 and ''s'' = 3 are called the dilogarithm (also referred to as Spence's function) and trilogarithm respectively. The name of the function comes from the fact that it may also be defined as the repeated integral of itself: :$$ \operatorname_(z) = \int_0^z \frac \,\mathrmt \,; thus the dilogarithm is an integral of the logarithm, and so on. For nonpositive integer orders ''s'', the polylogarithm is a rational function. ==Properties== Preliminary note: In the important case where the polylogarithm order $s$ is an integer, it will be represented by $n$ (or $-n$ when negative). It is often convenient to define $\backslash mu=\backslash ln(z)$ where $\backslash ln(z)$ is the principal branch of the complex logarithm $\backslash operatorname(\; z\; )$ so that . Also, all exponentiation will be assumed to be single-valued: $z^s\; =\; \backslash exp(\; s\; \backslash ln(\; z\; ))$. Depending on the order $s$, the polylogarithm may be multi-valued. The principal branch of $\backslash operatorname\_s(\; z)$ is taken to be given for by the above series definition and taken to be continuous except on the positive real axis, where a cut is made from $z\; =\; 1$ to $\backslash infty$ such that the axis is placed on the lower half plane of $z$. In terms of $\backslash mu$, this amounts to . The discontinuity of the polylogarithm in dependence on $\backslash mu$ can sometimes be confusing. For real argument $z$, the polylogarithm of real order $s$ is real if , and its imaginary part for $z\; \backslash ge\; 1$ is : :$\backslash textrm\backslash left(\; \backslash operatorname\_s(z)\; \backslash right)\; =\; -\backslash over\}\; \backslash ,.$ Going across the cut, if ''ε'' is an infinitesimally small positive real number, then: :$\backslash textrm\backslash left(\; \backslash operatorname\_s(z+i\backslash epsilon)\; \backslash right)\; =\; \backslash over\}\; \backslash ,.$ Both can be concluded from the series expansion (see below) of Li''s''(''e''''µ'') about ''µ'' = 0. The derivatives of the polylogarithm follow from the defining power series: :$z\; \backslash ,\; =\; \backslash operatorname\_(z)$ :$=\; \backslash operatorname\_(e^\backslash mu)\; \backslash ,.$ The square relationship is easily seen from the duplication formula (see also , ): :$$ \operatorname_s(-z) + \operatorname_s(z) = 2^ \,\operatorname_s(z^2) \,. Note that Kummer's function obeys a very similar duplication formula. This is a special case of the multiplication formula, for any positive integer ''p'': :$$ \sum_^\operatorname_s(z \,e^) = p^ \,\operatorname_s(z^p) \,, which can be proved using the series definition of the polylogarithm and the orthogonality of the exponential terms (see e.g. discrete Fourier transform). Another important property, the inversion formula, involves the Hurwitz zeta function or the Bernoulli polynomials and is found under relationship to other functions below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Polylogarithm」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.