https://mizugadro.mydns.jp/t/index.php?title=Maga&feed=atom&action=history Maga - Revision history 2020-09-27T07:48:53Z Revision history for this page on the wiki MediaWiki 1.31.10 https://mizugadro.mydns.jp/t/index.php?title=Maga&diff=28264&oldid=prev T: Text replacement - "\$([^\$]+)\$" to "\$$\1\$$" 2019-07-30T09:44:09Z <p>Text replacement - &quot;\$([^\$]+)\$&quot; to &quot;\$$\1\$$&quot;</p> <a href="https://mizugadro.mydns.jp/t/index.php?title=Maga&amp;diff=28264&amp;oldid=14646">Show changes</a> T https://mizugadro.mydns.jp/t/index.php?title=Maga&diff=14646&oldid=prev Maintenance script at 22:02, 30 November 2018 2018-11-30T22:02:38Z <p></p> <p><b>New page</b></p><div>[[File:MagaplotFragment.jpg|300px|thumb|$y\!=\!\mathrm{maga}(x)$ (thick black curve) and related functions]]<br /> <br /> Function [[maga]] appears at the consideration of the principal mode guided by the set of pinholes, and expresses the <br /> loss of the guiding.<br /> <br /> The scalar product of function [[mori]] to itself with quadratic defacement is denoted with identifier [[naga]]<br /> <br /> $\mathrm{naga}(x)=\displaystyle 2 \int_0^\infty \mathrm{mori}(p )^2 \exp(\mathrm i p^2 x) \, p \,\mathrm d p$<br /> <br /> Function [[mori]] is defined with <br /> <br /> [[mori]]$(p)=\displaystyle \frac{J_0(L p)}{1-p^2}$ <br /> <br /> and the corresponding limit at $p\!=\!\pm 1$; <br /> <br /> $L\!\approx\!2.404825557695773$ is the first zero of the [[Bessel function]] of 0th order, $J_0(L)\!=\!0$.<br /> <br /> The normalisation factor $2$ in the definition is chosen in such a way that $\mathrm{naga}(0)\!=\!1$.<br /> <br /> Then, for real values of the argument, [[maga]] is defined with<br /> <br /> $\mathrm{maga}(x)=1-|\mathrm{naga}(x)|^2$<br /> <br /> The explicit plot of function [[maga]] is shown with black curve in the figure at top.<br /> <br /> ==Properties==<br /> <br /> $\mathrm{maga}(-x)= \mathrm{maga}(x)^*$<br /> <br /> although the range of holomorphism gof this function is not yet well analysed,<br /> it is obvious that this function is singular at zero. For small real values of its argument, <br /> <br /> $\mathrm {maga}(x)= C_0\, x^{3/2} \big(1+c_1 x+O(x)^{3/2}\big)$<br /> <br /> where $C_0$ and $c_1$ are constants;<br /> <br /> $C_0\approx 0.44238$<br /> <br /> $c_1\approx - 0.085$<br /> <br /> Precise evaluation of constant $C_0$ requires either integration of oscillating function (that is slow, if performed in a straightforward way), or search for the appropriate contour integral in the complex plane. For the last option, the holomorphic properties of the functions involved (and their complex maps) should be analysed.<br /> <br /> At large real values of the argument, maga approaches unity; $~\displaystyle \lim_{x\rightarrow 0} \,\mathrm{maga}(x)=1$.<br /> <br /> ==Fourier==<br /> <br /> Integral in the definition of function[[Maga]] can be expressed through the Fourier transforms of function [[nori]] defined with<br /> <br /> [[nori]]$(x)=\,$[[kori]]$^2(x)=\,$[[mori]]$\big(\sqrt{x}\big)^2$.<br /> <br /> Function [[nori]] is shown in figure at the top with thin green line.<br /> Then, the change of variable of integration definition of function [[naga]] becomes straightforward. Let $q=p^2$; then, $\mathrm dq=2 p\mathrm dp$, and<br /> <br /> $\displaystyle \mathrm{naga}(x)=\int_0^\infty \mathrm{nori}(q) \, \exp(\mathrm i \,q\, x)\, \mathrm d q=\mathrm{nagc}(x) + \mathrm i ~\mathrm{nags}(x)$<br /> <br /> where <br /> <br /> [[nagc]]$(x)=\displaystyle \int_0^\infty \mathrm{nori}(q) \, \cos(q\, x)\, \mathrm d q$<br /> <br /> [[nags]]$(x)=\displaystyle \int_0^\infty \mathrm{nori}(q) \, \sin(q\, x)\, \mathrm d q$<br /> <br /> appear as non-normalised [[CosFT]] and [[SinFT]] of function [[nori]] for real argument $x$. These two functions are shown with thick blue and thick red lines in the figures at top.<br /> <br /> Then, function [[maga]] appears in the following way:<br /> <br /> $\mathrm {maga}(z)=1-\mathrm{nagc}(z)^2-\mathrm{nags}(z)^2$<br /> <br /> This representation indicates the way of holomorphic extension of function [[maga]] for complex values of the argument $z$; the appropriate paths of integration should be chosen in the integrals for [[nagc]]$(z)$ and [[nags]]$(z)$ in order to provide the convergence.<br /> <br /> As the straightforward evaluation of function [[maga]] through the [[SinFT]] and [[CosFT]] is not efficient, several service functions are suggested to boost the evaluation.<br /> <br /> ==Small argument==<br /> For application to the [[pinhole waveguide]], case of small argument is especially important.<br /> In order to boost evaluation in this case, define<br /> <br /> [[nag]]$(x)=\int_0^{\infty} \mathrm{kori}(x)^2 (1-\cos(q\,x)) \,\mathrm d q$<br /> <br /> Due to the unity norm of function [[kori]], the following relation takes place: [[nag]]$(x)=1-\,$[[nagc]]$(x)$, and<br /> <br /> [[nagc]]$(x)=1-\,$[[nag]]$(x)$<br /> <br /> Then,<br /> <br /> [[mori]]$(x)=1-\big(1-\mathrm{nag}(x)\big)^2-\mathrm{nags}(x)^2=2\,\mathrm{nag}(x)-\mathrm{nag}(x)^2-\mathrm{nags}(x)^2$<br /> <br /> The first term in the last expression determines the leading term in the asymptotic expansion of function [[mori]] at the small values of its argument.<br /> <br /> ==Fitting==<br /> The representation above can be used for evaluation and for the figging of function [[maga]].<br /> <br /> One of the simple fiting is provided with formula<br /> <br /> $\mathrm{fit2}(x)=\displaystyle \frac{x^{3/2}}{\sqrt{5-x/2+x^2/4+x^3}}$<br /> <br /> This fit is shown in the figure at top with dotted line. It is close to the straight curve $y=\mathrm{maga}(x)$. <br /> In order to see deviation of function fit2 from function maga, the difference, scaled with factor 50, is shown in the bottom part of the graphic.<br /> The difference oscillates, these oscillations become rapid and small at small values of the argument.<br /> <br /> Finction fit2 provides at least 2 significant figures of function [[maga]] and can be useful in the design of the [[pinhole waveguide]]s.<br /> &lt;ref&gt;http://www.academicjournals.org/app/webroot/ebook/journal1414510635_IJPS%20-%2030th%20Oct,%202014%20Issue.pdf#page=12 Makoto Morinaga. Guiding of light with pinholes. Physical Sciences ￼￼￼￼￼￼￼Volume 9 Number 20 30 October, 2014 ISSN 1992-1950, pages 444-453<br /> &lt;/ref&gt;<br /> <br /> ==Application==<br /> Function [[maga]] describes the first approximation loss in the [[pinhole waveguide]], in the limit of small loss. The argument $p$ of [[maga]]$(p )$ has the following meaning:<br /> <br /> $\displaystyle p= \frac{T}{2 k} \left(\frac{L_1}{r}\right)^2$<br /> <br /> where $k$ is wavenumber, $r$ is radius of the pinholes, and $T$ is distance between the pinholes. Then, function [[maga]] gives the first approximation in the loss of the [[Bessel mode]] in such a waveguide, in the limit of small $p$.<br /> <br /> The straightforward numerical integration in the definition of [[maga]] is not efficient, even for real $p$; for this reason, the analytical properties of the integrand, and, in particular, those of function [[kori]], deserve the detailed investigation, analysis and description.<br /> <br /> The goal is to calculate the asymptotic expansion of function [[maga]] at small values of its argument, ($\,|p| \!\ll\! 1\,$),<br /> at large values of its argument ($\,|p|\!\gg\! 1\,$), to check agreement between the approximations, based on these expansions, and to generate the [[complex map]] of function [[maga]]. For this goal, the fast and precise implementation of function [[kori]] is necessary.<br /> <br /> It may have sense to consider first the function [[naga]] defined as <br /> <br /> [[naga]]$(p )=\displaystyle<br /> \int_0^\infty \mathrm{kori}(x)^2\, \exp(\mathrm i p x) \, \mathrm d x$<br /> <br /> then $~$ [[maga]]$(z)=1-\mathrm{naga}(z)\, \mathrm{naga}(z^*)^*$<br /> <br /> Several articles about the functions introduced above are expected to be loaded in [[TORI]]. (Some of them ar already loaded, in particular, [[mori]], [[nori]] and [[kori]], but they are still far from the ideal description of a special function)<br /> Then it will be possible to choose the most efficient representation and description of function [[maga]], using some of these functions.<br /> <br /> The names [[kori]], [[nori]], [[mori]], [[maga]] and [[naga]], suggested for the functions involved, are tentative; they may be reconsidered and redefined for the analysis of the expansion of the mode of the [[pinhole waveguide]] with respect to the [[Bessel mode]]s and more accurate estimate of the loss in such a waveguide. <br /> <br /> In such a way, the descriptions and implementations of these functions appear as [[tool]]s for the description of the [[pinhole waveguide]], following the general ideology of [[TORI]]. Construction of these functions may look as a routine, dirty job (неблагодарная, не [[видная работа]]), that will not be appreciated by the scientific community, but, still, somebody is supposed to do it. Then, after a time, these functions may have also other applications. In particular, they may be useful for the &quot;ab initio&quot; calculus of the quantum reflection of excited atoms from a wall). Keeping in mind these applications, in have sense to perform the detailed analysis; to get the asymptotic expansions and the implementations with many decimal digits.<br /> <br /> ==References==<br /> &lt;references/&gt;<br /> <br /> ==Keywords==<br /> <br /> [[Bessel transform]],<br /> [[CosFT]]<br /> [[FFT]],<br /> [[Morinaga function]],<br /> [[Naga]],<br /> [[Nori]],<br /> [[SinFT]],<br /> <br /> [[Category:Bessel transform]]<br /> [[Category:English]]<br /> [[Category:Fourier transform]]<br /> [[Category:Maga]]<br /> [[Category:Makoto Morinaga]]<br /> [[Category:Mori]]<br /> [[Category:Naga]]<br /> [[Category:Nori]]<br /> [[Category:Paraxial approximation]]<br /> [[Category:Zeno effect]]</div> Maintenance script