Applets:Bessel Functions of the First Kind - Versionsgeschichte

Maintenance script: Add English interlanguage link

2026-03-16T12:38:46Z

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← Nächstältere Version		Version vom 16. März 2026, 14:38 Uhr
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	{{LntAppletLinkEn\|besselFuns_en}}		{{LntAppletLinkEn\|besselFuns_en}}
			[[en:Applets:Bessel_Functions_of_the_First_Kind]]

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2026-02-24T13:40:22Z

Join multi-line formulas for MW 1.43 parser compat

← Nächstältere Version		Version vom 24. Februar 2026, 15:40 Uhr
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	This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:		This applet allows the calculation and graphical representation of the Bessel functions of the first kind and $n$–th order according to the series representation:

	:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2		:$${\rm J}_n (x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^k \cdot (x/2)^{n \hspace{0.05cm} + \hspace{0.05cm} 2\hspace{0.05cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$
	\hspace{0.05cm}\cdot \hspace{0.05cm}k}}{k! \cdot (n+k)!} \hspace{0.05cm}.$$

	*The functions ${\rm J}_n (x)$ can be represented graphically for the order $n=0$ to $n=9$ in different colors.		*The functions ${\rm J}_n (x)$ can be represented graphically for the order $n=0$ to $n=9$ in different colors.
Zeile 20:		Zeile 19:
	Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form		Bessel functions (or cylinder functions) are solutions of the Bessel differential equation of the form

	:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2)		:$$x^2 \cdot \frac{ {\rm d}^2}{{\rm d}x^2}\ {\rm J}_n (x) \ + \ x \cdot \frac{ {\rm d}}{{\rm d}x}\ {\rm J}_n (x) \ + \ (x^2 - n^2) \cdot {\rm J}_n (x)= 0. $$
	\cdot {\rm J}_n (x)= 0. $$

	This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, also in this program. These mathematical functions, introduced by [https://en.wikipedia.org/wiki/Friedrich_Bessel Friedrich Wilhelm Bessel] in 1844, can also be represented in closed form as integrals:		This is an ordinary second-order linear differential equation. The parameter $ n $ is usually integer, also in this program. These mathematical functions, introduced by [https://en.wikipedia.org/wiki/Friedrich_Bessel Friedrich Wilhelm Bessel] in 1844, can also be represented in closed form as integrals:

	:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha		:$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.		The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.
Zeile 119:		Zeile 116:
	$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$		$$\hspace{0.2cm}x(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} x(t) +x_0 \hspace{0.05cm}, \hspace{0.2cm} y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} y(t) +y_0\hspace{0.05cm},$$

	$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}		$$\hspace{0.2cm}z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$
	z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.$$

	The graph shows the ''Rice–Fading channel model''. It can be summarized as follows:		The graph shows the ''Rice–Fading channel model''. It can be summarized as follows:
Zeile 129:		Zeile 125:

	*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the <i>modified Besselfunktion</i> zeroth order denotes:		*To simplify the notation, we set $\vert z(t)\vert = a(t)$. For $a < 0$ , the amounts - WDF are $f_a(a) \equiv 0$, for $a \ge 0$ the following equation holds, where ${\rm I_0}(x)$ is the <i>modified Besselfunktion</i> zeroth order denotes:
	:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =		:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ - (a^2 + \vert z_0 \vert ^2)/(2\sigma^2)} \cdot {\rm I}_0 \left [ \frac{a \cdot \vert z_0 \vert}{\sigma^2} \right ] \hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) = \sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}\hspace{0.05cm}.$$
	\sum_{k = 0}^{\infty} \frac{ (u/2)^{2k} }{k! \cdot \Gamma (k+1)}
	\hspace{0.05cm}.$$
	*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – in each case of first kind – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}		*Between the modified Bessel function and the traditional Bessel function ${\rm I_0}(x)$ – in each case of first kind – So the connection exists ${\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u)$.}}

Höfler: Textersetzung - „Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29“ durch „Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_von_1974-2024.29“

2025-02-24T09:43:01Z

Textersetzung - „Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29“ durch „Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_von_1974-2024.29“

← Nächstältere Version		Version vom 24. Februar 2025, 11:43 Uhr
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	==About the Authors==		==About the Authors==
	This interactive applet was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technische Universität München].		This interactive applet was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technische Universität München].
	*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29\|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29\|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.~~28am_LNT_seit_1974~~.29\|Günter Söder]]).		*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29\|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29\|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_von_1974-2024.29\|Günter Söder]]).
	*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29\|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_LÜT-Angehörige#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28bei_L.C3.9CT_von_2014-2022.29\|Tasnád Kernetzky]]).		*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29\|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_LÜT-Angehörige#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28bei_L.C3.9CT_von_2014-2022.29\|Tasnád Kernetzky]]).

Höfler am 26. Oktober 2023 um 10:05 Uhr

2023-10-26T10:05:34Z

← Nächstältere Version		Version vom 26. Oktober 2023, 12:05 Uhr
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	This interactive applet was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technische Universität München].		This interactive applet was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technische Universität München].
	*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29\|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29\|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29\|Günter Söder]]).		*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29\|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29\|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29\|Günter Söder]]).
	*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29\|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/~~Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik~~#~~Tasn.C3~~.~~A1d_Kernetzky~~.~~2C_M~~.Sc._.28bei_L.C3.~~9CT_seit_2014~~.29\|Tasnád Kernetzky]]).		*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29\|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_LÜT-Angehörige#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28bei_L.C3.9CT_von_2014-2022.29\|Tasnád Kernetzky]]).

	==Once again: Open Applet in new Tab==		==Once again: Open Applet in new Tab==

	{{LntAppletLinkEn\|besselFuns_en}}		{{LntAppletLinkEn\|besselFuns_en}}

Tasnad am 10. Juni 2020 um 08:04 Uhr

2020-06-10T08:04:17Z

← Nächstältere Version		Version vom 10. Juni 2020, 10:04 Uhr
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	[[Applets:~~Besselfunktionen_erster_Art_(neues_Applet)~~\|'''German description''']]		[[Applets:Besselfunktionen_erster_Art\|'''German description''']]

	==Theoretical Background==		==Theoretical Background==

Tasnad am 9. Juni 2020 um 20:46 Uhr

2020-06-09T20:46:53Z

← Nächstältere Version		Version vom 9. Juni 2020, 22:46 Uhr
Zeile 1:		Zeile 1:
	{{LntAppletLinkEn\|~~bessel~~}}		{{LntAppletLinkEn\|besselFuns_en}}

	==Applet Description==		==Applet Description==
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	==Once again: Open Applet in new Tab==		==Once again: Open Applet in new Tab==

	{{LntAppletLinkEn\|~~bessel~~}}		{{LntAppletLinkEn\|besselFuns_en}}

Guenter am 27. August 2018 um 16:31 Uhr

2018-08-27T16:31:51Z

← Nächstältere Version		Version vom 27. August 2018, 18:31 Uhr
Zeile 1:		Zeile 1:
	{{~~LntAppletLink~~\|bessel}}		{{LntAppletLinkEn\|bessel}}

	==Applet Description==		==Applet Description==
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	*Also in the frequency modulation, the Bessel lines now occur at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there are now more significantly more Bessel lines at the bottom due to the larger modulation index $η = 2.5$ than in the upper right (for $η = 1.5$ valid) chart.}}		*Also in the frequency modulation, the Bessel lines now occur at a distance of $3 \ \rm kHz$. However, since FM has a modulation index inversely proportional to $f_{\rm N}$, there are now more significantly more Bessel lines at the bottom due to the larger modulation index $η = 2.5$ than in the upper right (for $η = 1.5$ valid) chart.}}


	~~'''Das folgende Kapitel muss noch angepasst werden!'''~~

	==Applet Manual==		==Applet Manual==
Zeile 185:		Zeile 183:
	*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29\|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29\|Tasnád Kernetzky]]).		*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29\|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29\|Tasnád Kernetzky]]).

	==~~Nochmalige Aufrufmöglichkeit des Applets~~ in ~~neuem Fenster~~==		==Once again: Open Applet in new Tab==

	{{~~LntAppletLink~~\|bessel}}		{{LntAppletLinkEn\|bessel}}

Guenter am 27. August 2018 um 16:28 Uhr

2018-08-27T16:28:52Z

← Nächstältere Version		Version vom 27. August 2018, 18:28 Uhr
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	==About the Authors==		==About the Authors==
	This interactive ~~calculation~~ was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ ~~Technical University of Munich~~] .		This interactive applet was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technische Universität München].
	*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29\|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29\|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29\|Günter Söder]]).		*The original version was created in 2005 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29\|Markus Elsberger]] and [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29\|Slim Lamine]] as part of thesis using &bdquo;FlashMX–Actionscript” (Supervisor: [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29\|Günter Söder]]).
	*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29\|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29\|Tasnád Kernetzky]]).		*In 2018 this Applet was redesigned and updated to &bdquo;HTML5” by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29\|Xiaohan Liu]] as part of her Bachelor's thesis (Supervisor: [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29\|Tasnád Kernetzky]]).

Guenter am 27. August 2018 um 16:22 Uhr

2018-08-27T16:22:45Z

← Nächstältere Version		Version vom 27. August 2018, 18:22 Uhr
Zeile 57:		Zeile 57:
	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example (B):}$   For the spectrum of the analytic signal, the following applies to ''Phase Modulation of a sinusoidal signal'':		$\text{Example (B):}$   For the spectrum of the analytic signal, the following applies to ''Phase Modulation of a sinusoidal signal'':
	[[Datei:Mod_T_3_1_S4_version2.png\|right\|frame\|Spectrum of the ~~analytic signal in phase modulation~~]]		[[Datei:Mod_T_3_1_S4_version2.png\|right\|frame\|Phase Modulation:   Spectrum of the Analytic Signal]]
	:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$		:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
	Denote this		Denote this
	*$f_{\rm T}$ the carrier frequency,		*$f_{\rm T}$ the carrier frequency   (German:   $\rm T$rägerfrequenz),
	*$f_{\rm N}$ the ~~message~~ frequency,		*$f_{\rm N}$ the frequency of the source signal   (German:   $\rm N$achrichtenfrequenz),
	* $A_{\rm T}$ the carrier amplitude.		* $A_{\rm T}$ the carrier amplitude   (German:   $\rm T$rägeramplitude).


Zeile 68:		Zeile 68:

	Based on the graphic, the following statements are possible:		Based on the graphic, the following statements are possible:
	*$S_+(f)$ consists here of ~~infinitely many~~ discrete lines at a distance of $f_{\rm N}$.		*$S_+(f)$ consists here of an infinite number of discrete lines at a distance of $f_{\rm N}$.
	*It is ~~thus~~ in principle infinitely extended.		*Thus the sectrum is in principle infinitely extended.
	*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.		*The weights of the spectral lines at $f_{\rm T} + n · f_{\rm N}$ ($n$ integer) are determined by the modulation index $η$ over the Bessel functions ${\rm J}_n(η)$.
	*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetrical about $f_{\rm T}$ for even $n$.		*The spectral lines are real in the case of a sinusoidal source signal and a cosinusoidal carrier and symmetrical about $f_{\rm T}$ for even $n$.
	*In the case of odd $n$, a change of sign corresponding to $\text{Property (B)}$ must be taken into account.		*In the case of odd $n$, a change of sign corresponding to $\text{Property (B)}$ must be taken into account.
	*The ~~phase modulation~~ of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}		*The Phase Modulation of another phase oscillation of source and / or carrier signal provides the same magnitude spectrum.}}
	<br><br>		<br><br>

Zeile 79:		Zeile 79:

	The applications of the Bessel functions in nature and engineering are diverse and play an important role in physics, for example:		The applications of the Bessel functions in nature and engineering are diverse and play an important role in physics, for example:
	* ~~Investigation of natural vibrations of~~ cylindrical ~~resonators~~,		* Electromagnetic waves in a cylindrical waveguide,
	* ~~Solution of~~ the radial Schrödinger-equation,		* Solutions to the radial Schrödinger equation,
	* ~~Sound pressure~~ amplitudes of ~~low-viscosity~~ rotational flows,		* Pressure amplitudes of inviscid rotational flows,
	* Heat conduction in cylindrical ~~bodies~~,		* Heat conduction in a cylindrical object,
	* ~~Scattering problem of~~ a ~~grid~~,		* Diffusion problems on a lattice,
	* Dynamics of ~~oscillating~~ bodies,		* Dynamics of floating bodies,
			*Frequency-dependent friction in circular pipelines
	* Angular resolution.		* Angular resolution.


	The Bessel functions are counted among the special functions because of their many applications in mathematical physics.		The Bessel functions are counted among the special functions because of their many applications in mathematical physics.

	In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.		In the following, we restrict ourselves to a few areas that are addressed in our tutorial $\rm LNTwww$.

	~~'''In the English original'''~~
	~~Electromagnetic waves in a cylindrical waveguide~~
	~~Pressure amplitudes of inviscid rotational flows~~
	~~Heat conduction in a cylindrical object~~
	~~Modes of vibration of a thin circular (or annular) acoustic membrane (such as a drum or other membranophone)~~
	~~Diffusion problems on a lattice~~
	~~Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle~~
	~~Solving for patterns of acoustical radiation~~
	~~Frequency-dependent friction in circular pipelines~~
	~~Dynamics of floating bodies~~
	~~Angular resolution~~
	~~'''The Ende'''~~

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Guenter am 27. August 2018 um 15:50 Uhr

2018-08-27T15:50:15Z

← Nächstältere Version		Version vom 27. August 2018, 17:50 Uhr
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	The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.		The functions ${\rm J}_n (x)$ belong to the class of Bessel functions of the first kind (German:   ''Besselfunktionen erster Art''). The parameter $n$ is called the ''Order''.

	''Annotation:''   There are a number of modifications of the Bessel functions, including the second kind ~~Bessel functions~~ named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$~~–~~functions. However, in this applet, only the first kind Bessel functions ~~are   ⇒  ~~ ${\rm J}_n (x)$ is considered.		''Annotation:''   There are a number of modifications of the Bessel functions, including the Bessel functions of the second kind named ${\rm Y}_n (x)$. For integer $n$, ${\rm Y}_n (x)$ can be replaced by ${\rm J}_n (x)$ functions. However, in this applet, only the first kind Bessel functions ${\rm J}_n (x)$ are considered.
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	$\text{Example (B):}$   For the spectrum of the analytic signal, the following applies to ~~[[Modulationsverfahren/Phasenmodulation_(PM)#Spektralfunktion_eines_phasenmodulierten_Sinussignals\|phase modulation~~ of a sinusoidal signal]]:		$\text{Example (B):}$   For the spectrum of the analytic signal, the following applies to ''Phase Modulation of a sinusoidal signal'':
	[[Datei:Mod_T_3_1_S4_version2.png\|right\|frame\|Spectrum of the analytic signal in phase modulation]]		[[Datei:Mod_T_3_1_S4_version2.png\|right\|frame\|Spectrum of the analytic signal in phase modulation]]
	:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$		:$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$