|
Coshc Function
In mathematics, the coshc function appears frequently in papers about optical scattering, Heisenberg spacetime and hyperbolic geometry. For z \neq 0, it is defined as \operatorname(z)=\frac 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\operatorname z \approx \left(z^+\frac z+\frac z^3+\frac z^5+\frac z^7+\frac z^9+\frac z^+\frac z^+O(z^) \right) The Padé approximant is\operatorname \left( z \right) = In terms of other special functions * \operatorname(z) = \frac , where (a,b,z) is Kummer's confluent hypergeometric function. *\operatorname(z)=\frac\,\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 * Tanc function * Tanhc function * Sinhc function References [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Cardinal Hyperbolic Cosine Function Coshc(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 already 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 pronouns 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 followed by a consonant s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
Optical Scattering
Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiation) in the medium through which they pass. In conventional use, this also includes deviation of reflected radiation from the angle predicted by the law of reflection. Reflections of radiation that undergo scattering are often called ''diffuse reflections'' and unscattered reflections are called ''specular'' (mirror-like) reflections. Originally, the term was confined to light scattering (going back at least as far as Isaac Newton in the 17th century). As more "ray"-like phenomena were discovered, the idea of scattering was extended to them, so that William Herschel could refer to the scattering of "heat rays" (not then recognized as electromagnetic in nature) in 1800. John Tyndall, a pioneer in light scattering researc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.png "Click for more on -> The Cardinal Hyperbolic Cosine Function Coshc(z) Plotted In The Complex Plane From -2-2i To 2+2i")
