{"id":306,"date":"2020-08-05T04:20:25","date_gmt":"2020-08-05T04:20:25","guid":{"rendered":"http:\/\/soulofmathematics.com\/?p=306"},"modified":"2020-08-05T05:47:07","modified_gmt":"2020-08-05T05:47:07","slug":"bessels-function","status":"publish","type":"post","link":"https:\/\/soulofmathematics.com\/index.php\/bessels-function\/","title":{"rendered":"Bessel’s Function"},"content":{"rendered":"\n

Just imagine a dew drop falling from a leaf on the surface of an absolutely calm lake, creating short ripples in water which then fight among themselves and disappear in time. Such an amazing thing to watch but, today we will unfold the mathematics behind it.<\/em><\/p>\n\n\n\n

The function which defines the phenomenon stated above or any acoustic activity on a circular membrane (exploited by most of the musical instruments) is called the Bessel’s function in the honour of the German astronomer Friedrich Wilhelm Bessel<\/a><\/strong>.<\/p>\n\n\n\n

The Bessel function was first defined by famous mathematician Daniel Bernoulli<\/a> and then generalized by  Friedrich Bessel during an investigation of solutions of one of Kepler’s equations of planetary motion. <\/p>\n\n\n\n

Bessel functions are used to solve in 3-D the wave equation at a given (harmonic) frequency. The solution is generally a sum of spherical Bessel’s functions that gives the acoustic pressure at a given location of the 3-D space.<\/p>\n\n\n\n

The Bessel’s function is nothing but the canonical solutions of the Bessel’s differential equation- <\/p>\n\n\n\n

\"{\\displaystyle<\/figure>\n\n\n\n

for an arbitrary complex number<\/a> \u03b1<\/em>. Although \u03b1<\/em> and \u2212\u03b1<\/em> produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \u03b1<\/em>. The most important cases are when \u03b1<\/em> is an integer or half-integer.<\/p>\n\n\n\n

The Bessel’s function can be categorized as follows-<\/p>\n\n\n\n

TYPE<\/th>FIRST KIND<\/th>SECOND KIND<\/th><\/tr><\/thead>
Bessel functions<\/td>J\u03b1<\/sub><\/td>Y\u03b1<\/sub><\/td><\/tr>
Modified Bessel functions<\/td>I\u03b1<\/sub><\/td>K\u03b1<\/sub><\/td><\/tr>
Hankel functions<\/td>H(1)<\/sup>\u03b1<\/sub> = J\u03b1<\/sub> + iY\u03b1<\/sub><\/td>H(2)<\/sup>\u03b1<\/sub> = J\u03b1<\/sub> \u2212 iY\u03b1<\/sub><\/td><\/tr>
Spherical Bessel functions<\/td>jn<\/sub><\/td>yn<\/sub><\/td><\/tr>
Spherical Hankel functions<\/td>h(1)<\/sup>n<\/sub> = jn<\/sub> + iyn<\/sub><\/td>h(2)<\/sup>n<\/sub> = jn<\/sub> \u2212 iyn<\/sub><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Bessel functions of the first kind of order n and -n<\/strong> –<\/p>\n\n\n\n

We define the Bessel functions of the first kind as-<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"
Graph of Bessel’s Function of First Kind<\/figcaption><\/figure>\n\n\n\n

Bessel’s Function has a wide range of application in electromagnetic wave theory, solution to the radial Schr\u00f6dinger equation<\/a> (in spherical and cylindrical coordinates) for a free particle and many more. We will try understand few of them in the days to come.<\/h4>\n\n\n\n
\n
All Posts<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Just imagine a dew drop falling from a leaf on the surface of an absolutely calm lake, creating short ripples in water which then fight among themselves and disappear in time. Such an amazing thing to watch but, today we will unfold the mathematics behind it. The function which defines the phenomenon stated above or any acoustic activity on a circular membrane (exploited by most of the musical instruments) is called the Bessel’s function in the honour of the German astronomer Friedrich Wilhelm Bessel. The Bessel function was first defined by famous mathematician Daniel Bernoulli and then generalized by  Friedrich Bessel during an investigation of solutions of one of Kepler’s equations of planetary motion. Bessel functions are used to solve in 3-D the wave equation at a given (harmonic) frequency. The solution is generally a sum of spherical Bessel’s functions that gives the acoustic pressure at a given location of the 3-D space. The Bessel’s function is nothing but the canonical solutions of the Bessel’s differential equation- for an arbitrary complex number \u03b1. Although \u03b1 and \u2212\u03b1 produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \u03b1. The most important cases are when \u03b1 is an integer or half-integer. The Bessel’s function can be categorized as follows- TYPE FIRST KIND SECOND KIND Bessel functions J\u03b1 Y\u03b1 Modified Bessel functions I\u03b1 K\u03b1 Hankel functions H(1)\u03b1 = J\u03b1 + iY\u03b1 H(2)\u03b1 = J\u03b1 \u2212 iY\u03b1 Spherical Bessel functions jn yn Spherical Hankel functions h(1)n = jn + iyn h(2)n = jn \u2212 iyn Bessel functions of the first kind of order n and -n – We define the Bessel functions of the first kind as- Bessel’s Function has a wide range of application in electromagnetic wave theory, solution to the radial Schr\u00f6dinger equation (in spherical and cylindrical coordinates) for a free particle and many more. We will try understand few of them in the days to come.<\/p>\n","protected":false},"author":1,"featured_media":307,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-306","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-sneak-peeks"],"featured_image_src":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/tumblr_mzxa8rG9BR1r2geqjo1_500.gif?fit=500%2C500&ssl=1","blog_images":{"medium":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/tumblr_mzxa8rG9BR1r2geqjo1_500.gif?fit=300%2C300&ssl=1","large":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/tumblr_mzxa8rG9BR1r2geqjo1_500.gif?fit=500%2C500&ssl=1"},"ams_acf":[],"yoast_head":"\nBessel's Function - SOUL OF MATHEMATICS<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/soulofmathematics.com\/index.php\/bessels-function\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Bessel's Function - SOUL OF MATHEMATICS\" \/>\n<meta property=\"og:description\" content=\"Just imagine a dew drop falling from a leaf on the surface of an absolutely calm lake, creating short ripples in water which then fight among themselves and disappear in time. Such an amazing thing to watch but, today we will unfold the mathematics behind it. The function which defines the phenomenon stated above or any acoustic activity on a circular membrane (exploited by most of the musical instruments) is called the Bessel’s function in the honour of the German astronomer Friedrich Wilhelm Bessel. The Bessel function was first defined by famous mathematician Daniel Bernoulli and then generalized by  Friedrich Bessel during an investigation of solutions of one of Kepler’s equations of planetary motion. Bessel functions are used to solve in 3-D the wave equation at a given (harmonic) frequency. The solution is generally a sum of spherical Bessel’s functions that gives the acoustic pressure at a given location of the 3-D space. The Bessel’s function is nothing but the canonical solutions of the Bessel’s differential equation- for an arbitrary complex number \u03b1. Although \u03b1 and \u2212\u03b1 produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \u03b1. The most important cases are when \u03b1 is an integer or half-integer. The Bessel’s function can be categorized as follows- TYPE FIRST KIND SECOND KIND Bessel functions J\u03b1 Y\u03b1 Modified Bessel functions I\u03b1 K\u03b1 Hankel functions H(1)\u03b1 = J\u03b1 + iY\u03b1 H(2)\u03b1 = J\u03b1 \u2212 iY\u03b1 Spherical Bessel functions jn yn Spherical Hankel functions h(1)n = jn + iyn h(2)n = jn \u2212 iyn Bessel functions of the first kind of order n and -n – We define the Bessel functions of the first kind as- Bessel’s Function has a wide range of application in electromagnetic wave theory, solution to the radial Schr\u00f6dinger equation (in spherical and cylindrical coordinates) for a free particle and many more. 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Such an amazing thing to watch but, today we will unfold the mathematics behind it. The function which defines the phenomenon stated above or any acoustic activity on a circular membrane (exploited by most of the musical instruments) is called the Bessel’s function in the honour of the German astronomer Friedrich Wilhelm Bessel. The Bessel function was first defined by famous mathematician Daniel Bernoulli and then generalized by  Friedrich Bessel during an investigation of solutions of one of Kepler’s equations of planetary motion. Bessel functions are used to solve in 3-D the wave equation at a given (harmonic) frequency. The solution is generally a sum of spherical Bessel’s functions that gives the acoustic pressure at a given location of the 3-D space. The Bessel’s function is nothing but the canonical solutions of the Bessel’s differential equation- for an arbitrary complex number \u03b1. Although \u03b1 and \u2212\u03b1 produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \u03b1. The most important cases are when \u03b1 is an integer or half-integer. The Bessel’s function can be categorized as follows- TYPE FIRST KIND SECOND KIND Bessel functions J\u03b1 Y\u03b1 Modified Bessel functions I\u03b1 K\u03b1 Hankel functions H(1)\u03b1 = J\u03b1 + iY\u03b1 H(2)\u03b1 = J\u03b1 \u2212 iY\u03b1 Spherical Bessel functions jn yn Spherical Hankel functions h(1)n = jn + iyn h(2)n = jn \u2212 iyn Bessel functions of the first kind of order n and -n – We define the Bessel functions of the first kind as- Bessel’s Function has a wide range of application in electromagnetic wave theory, solution to the radial Schr\u00f6dinger equation (in spherical and cylindrical coordinates) for a free particle and many more. 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