Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0002 360856 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.1126 1209600 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.1126 1209600 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.2170 4190088 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.2184 4217848 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.2192 4235560 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.2194 4266080 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.2204 4288800 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.2205 4288800 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2711 4970720 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2749 5162872 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2755 5183800 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2879 5385264 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2879 5385520 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Legendre's Equation - Legendre Polynomials

The second order, non constant coefficient, differential equationwith boundary conditionis finite is known as Legendre's equation. Solutions exist forand the normalized solutions,are polynomials of degreeknown as Legendre polynomials. These polynomials are defined only on the intervalbecause ator 1 the coefficient ofis zero.

The first few Legendre polynomials are given by

Legendre polynomials form a mutually orthonormal set, that iswhere

Proof: The P-n (x) satisfy

We can write this as(1)

Similarly, for a solutionwe can write(2)

Multiply (1) byand integrate between -1 and 1:

Integrating the first term by parts gives

The first term on the right is zero:

Similarly for (2)

The right hand side of both equations are the same so

Sincewe must have both sides are zero hence

Add comment

Security code
Refresh