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{{Short description|Theorem on rational values of the sine}} ...0°}} for which the [[sine]] of ''{{Mvar|&theta;}}''&nbsp;degrees is also a rational number are:<ref>{{cite journal |last=Schaumberger |first=Norman |year=1974 ...

...] of continuous functions on compact subsets of the [[complex plane]] by [[rational function]]s. The theorem was proved in 1931 by the German mathematicians [[ ...complex-valued function on ''K'' can be uniformly approximated by rational functions. ...

...) of rational functions (which are natural generalizations for meromorphic functions in the non-complex realm) having a zero or a pole at ''P''. ...al ring at a point ''P'' is actually conformed by the germs of holomorphic functions vanishing at ''P''. ...

...s on ''Y'' may be viewed as invariant rational functions on ''X'' (i.e., [[rational-invariant]]s of ''X''). ...

...(probability theory)|characteristic function]]s that are strictly proper [[rational function]]s (i.e., the degree of the numerator is strictly less than the de ...obability density functions are exponential-polynomial-trigonometric (EPT) functions. ...

...sets and is closed under [[set union|union]], product and [[Kleene star]]. Rational sets are useful in [[automata theory]], [[formal language]]s and [[algebra] A rational set generalizes the notion of [[regular language|rational (or regular) language]] (understood as defined by [[regular expression]]s) ...

...ree <math>\le n</math> and denominator has degree <math>\le m</math>, the rational function <math>g = p/q</math>, with <math>p</math> and <math>q</math> being ...

...=Jones|first1=Rafe|last2=Levy|first2=Alon|title=Eventually stable rational functions|journal=International Journal of Number Theory|volume=13|number=9|pages=229 ==Generalization to rational functions and arbitrary basepoints== ...

...on ''H'' of functions that are defined for all large real numbers, that is functions ''f'' that map (''u'',∞) to the real numbers '''R''', for some real number ...an | first1=Michael | title=Hardy fields and existence of transexponential functions | year=1986 | journal=[[Aequationes Mathematicae]] | doi=10.1007/BF02189932 ...

...ions|complex functions]], ''Z''(''s''), of a complex variable, ''s''. A [[rational function]] is defined to have the PR property if it has a positive real par ...his sense means that the impedance can be constructed from a finite (hence rational) number of discrete ideal [[Passivity (engineering)|passive]] linear elemen ...

{{DISPLAYTITLE:Special values of ''L''-functions}} In [[mathematics]], the study of '''special values of {{mvar|L}}-functions''' is a subfield of [[number theory]] devoted to generalising formulae such ...

{{short description|Method of integration for rational functions}} ...rt{ax^2 + bx + c}</math>. In such cases, the integrand can be changed to a rational function by using the substitutions of Euler.<ref>N. Piskunov, ''Diferentsi ...

...ield of fractions]] of a convolution [[ring (abstract algebra) | ring]] of functions: a convolution quotient is to the [[operation (mathematics)|operation]] of ...fine convolution quotients by saying that for two [[function (mathematics)|functions]] ''ƒ'',&nbsp;''g'', the pair (''ƒ'',&nbsp;''g'') has the same convolution ...

[[Image:Herman Standard.png|200px|right|thumb|The Julia set of the cubic rational function ''e''^2&pi;''it''''z''²(''z''&minus;4)/(1&min ...tudies, 160, Princeton Univ. Press, Princeton, NJ, 2006.</ref> where the [[rational function]] is conformally conjugate to an [[irrational rotation]] of the st ...

...correspond to the symmetries and periodicities of the basic trigonometric functions. ...ac{P(\sin t, \cos t)}{Q(\sin t, \cos t)}</math>) to the integration of a [[rational function]] in a new variable, which can be calculated by [[partial fraction ...

...R models, which rely on backward-looking assumptions, RE-SVAR incorporates rational expectations (RE) explicitly. ''E_t'': Represents the [[rational expectations]] operator at time ''t'', ...

...)|surface]]s, methods have been developed for [[computer-aided design]] of rational motions. ...d that [[Bézier curve|rational Bézier]] and [[Nonuniform rational B-spline|rational B-spline]] based curve representation schemes can be combined with [[dual q ...

The fields of [[rational number]]s and [[real number]]s, with their usual orderings, satisfy the Arc ...he [[surreal number]]s, the [[Dehn planes|Dehn field]], and the field of [[rational function]]s with real coefficients (where we define {{math|''f'' > ''g''}} ...

...>(''τ'',''0'') of a [[theta function]] ''θ''_''m''(τ,''z'') with rational characteristic ''m'' to ''z''&nbsp;=&nbsp;0. The variable ''τ'' may be a co ...,''b''</sub>(''τ'',''z'') are the four [[Jacobi theta functions]], and the functions θ_''a'',''b''(''τ'',0) are the classical Jacobi theta constants. ...

...rate any [[rational fraction|rational expression]] involving trigonometric functions.<ref>{{Cite journal|last=Kilburn|first=Korey|title=Applying Euler's Formula ...an integrate the complex exponentials and not change back to trigonometric functions until the end: ...