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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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]   |
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Sinhc Integral
In mathematics, trigonometric integrals are a family of integrals involving trigonometric functions. Sine integral The different sine integral definitions are \operatorname(x) = \int_0^x\frac\,dt \operatorname(x) = -\int_x^\infty\frac\,dt~. Note that the integrand is the sinc function, and also the zeroth spherical Bessel function. Since is an even entire function (holomorphic over the entire complex plane), is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints. By definition, is the antiderivative of whose value is zero at , and is the antiderivative whose value is zero at . Their difference is given by the Dirichlet integral, \operatorname(x) - \operatorname(x) = \int_0^\infty\frac\,dt = \frac \quad \text \quad \operatorname(x) = \frac + \operatorname(x) ~. In signal processing, the oscillations of the sine integral cause overshoot and ringing artifacts when using the sinc filter, and frequency domain ringing ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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]   |
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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]   |
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Sinhc Re Plot
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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Sinhc Im Plot
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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Sinhc Abs Plot
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]   |