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Tanhc Function
In mathematics, the tanhc function is defined for z \neq 0 as \operatorname(z)=\frac The tanhc function is the hyperbolic analogue of the tanc function. Properties The first-order derivative is given by : \frac - \frac The Taylor series expansion\operatorname z \approx \left(1-\frac z^2 + \frac z^4 - \frac z^6 + \frac z^8 - \frac z^ + \frac z^ - \frac z^+O(z^) \right)which leads to the series expansion of the integral as\int _^\!=(z-^+^-^+^-^+O \left( ^ \right) ) The Padé approximant is \operatorname \left( z \right) = \left( 1+\,^+\,^+\,^+\,^ \right) \left( 1+\,^+\,^+\,^+\,^ \right) ^ In terms of other special functions * \operatorname(z)=2\,, where (a,b,z) is Kummer's confluent hypergeometric function. *\operatorname(z)=2 \frac , where (q, \alpha, \gamma, \delta, \epsilon ,z) is the biconfluent Heun function. * \operatorname(z)= \frac z, where (a,b,z) is a Whittaker function. Gallery {, , , , See also * Sinhc function * Tanc function * Coshc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Cardinal Hyperbolic Tangent Function Tanhc(z) Plotted In The Complex Plane From -2-2i To 2+2i
''The'' () is a grammatical Article (grammar), article in English language, English, denoting persons or things that are already or about to be mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the Most common words in English, most frequently used word in the English language; studies and analyses of texts have found it to account for seven percent of all printed English-language words. It is derived from gendered articles in Old English which combined in Middle English and now has a single form used with nouns of any gender. The word can be used with both singular and plural nouns, and with a noun that starts with any letter. This is different from many other languages, which have different forms of the definite article for different genders or numbers. Pronunciation In most dialects, "the" is pronounced as (with the voiced dental fricative followed by a schwa) when fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tanc Function
In mathematics, the tanc function is defined for z \neq 0 as \operatorname(z)=\frac Properties The first-order derivative of the tanc function is given by : \frac - \frac The Taylor series expansion is\operatorname z \approx \left(1+ \frac z^2 + \frac z^4 + \frac z^6 + \frac z^8 + \frac z^ + \frac z^+ \frac z^ + O(z^ ) \right)which leads to the series expansion of the integral as\int _0^z \frac \, dx = \left(z+ \frac z^3 + \frac z^5 + \frac z^7 + \frac z^9+ \frac z^+ \frac z^ + \frac z^+ O (z^) \right)The Padé approximant is\operatorname \left( z \right) = \left( 1-\,^ + \,^-\,^+\,^ \right) \left( 1-\,^+\,^-\,^+\,^ \right) ^ In terms of other special functions * \operatorname(z)=, where (a,b,z) is Kummer's confluent hypergeometric function. *\operatorname(z)= \frac , where (q, \alpha, \gamma, \delta, \epsilon ,z) is the biconfluent Heun function. * \operatorname(z)= \frac , where (a,b,z) is a Whittaker function. Gallery {, , , , See also * Sinhc f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sinhc Function
In mathematics, the sinhc function appears frequently in papers about optical scattering, Heisenberg spacetime and hyperbolic geometry. For z \neq 0, it is defined as \operatorname(z)=\frac The sinhc function is the hyperbolic analogue of the sinc function, defined by \sin x/x. It is a solution of the following differential equation: w(z) z-2\,\frac w (z) -z \frac w (z) =0 Properties The first-order derivative is given by : \frac - \frac The Taylor series expansion is\sum_^\infty \frac.The Padé approximant is \operatorname \left( z \right) = \left( 1+ \,^+\,^+\,^+\,^ \right) \left( 1-\,^+\,^-\,^ +\,^ \right) ^ In terms of other special functions * \operatorname(z)=\frac , where (a,b,z) is Kummer's confluent hypergeometric function. * \operatorname(z)=\frac , where (q, \alpha, \gamma, \delta, \epsilon ,z) is the biconfluent Heun function. * \operatorname(z)=1/2\,\frac , where (a,b,z) is a Whittaker function. Gallery {, , , , See also *Tanc function *Ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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