# polynôme de legendre

ℓ are orthogonal, parameterized by θ over ( λ ≥ ⁡ , μ ⁡ ⁡ *���o��$�N �~B�0x&������@�o�E��X㪼;���;�A� 2v{&D!��Ӊ���r-4̵ق.o,�A�w�S;V��~��S3�D32�����}|��GR �e��Boy�#ݟq2O�_��s6hj�[email protected]~k[z��)���c�5R�)O8�alHڟ9���A�� Z���y� 䄑�\�˴��0�gTpE�����ۃϲ�8�2H���6'iH�-�;���d7��~�1����S/)&�m��]y%��}�B[�1k|�N��1M�xR��+M#5�����pZT��KYl�����2)� ��>��gD}�>~.��g�>~*�SӬ2�wk�ʛ�5�Ue��2tm۾M�GC��Z��ejf˭͢|CW����$P�x�6�/ֆ\�g�!��kC�u�Ǘy�Q�֔?�(WH������ �$�D�dR��''��|��^3���F>c��:1��2��}=n)?���Nn�Ŧ���AV8A����!I86K�(5ML��d���A�E��0�>���.=��F^�Ņ�7��p���[\���l�wκ)�p�!�1]�����N��Vb��Q�;Q�1���ۈ4рI���RM�� �����W��$d&��2�K��� ��� �[��Ł�[+1�G�0W�>�09ʂh(a�<9��O���[�4��,����W��ǘgЏ��ZL܁wG��V�:a�qɓ1/�4yI. and those solutions are proportional to. ⁡ sin The associated Legendre polynomials are not mutually orthogonal in general. 3 0 obj << λ 2 LegendreP [n, m, a, z] gives Legendre functions of type a. The Legendre ordinary differential equation is frequently encountered in physics and other technical fields. �����7�l��A��ɦ�2�2�q� �8��)= for integer m≥0, and an equation for the θ-dependent part. for fixed m, cos 0. or Dong and Lemus (2002) generalized the derivation of this formula to integrals over a product of an arbitrary number of associated Legendre polynomials. ℓ ⁡ The solutions are usually written in terms of complex exponentials: The functions is not orthogonal to ℓ By repeating the argument, we get cn+4 = 0 and in general cn+2k = 0 for k ≥ 1. P ) P {\displaystyle \ell {\geq }m} ϕ The colatitude angle in spherical coordinates is {\displaystyle \lambda =\ell (\ell +1)\,} In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates. the same differential equation as before: Since this is a second order differential equation, it has a second solution, , the list given above yields the first few polynomials, parameterized this way, as: The orthogonality relations given above become in this formulation: ( {\displaystyle (1-x^{2})^{1/2}=\sin \theta } m = {\displaystyle _{2}F_{1}} Ääntämisohje: Opi, kuinka äännetään sana polynôme de Legendre äidinkielen tasoisesti kielellä ranska. This formula is to be used under the following assumptions: Other quantities appearing in the formula are defined as. = Γ P ℓ cos {\displaystyle P_{\ell }^{m}(\cos \theta )} �[���HU}UT�s�P������V�KQ�7V��+���T��>�М��鋸��i�>=5 , defined as: P ) <> The functions described by this equation satisfy the general Legendre differential equation with the indicated values of the parameters ℓ and m follows by differentiating m times the Legendre equation for Pℓ:, This equation allows extension of the range of m to: −ℓ ≤ m ≤ ℓ. The first few associated Legendre functions, including those for negative values of m, are: These functions have a number of recurrence properties: Helpful identities (initial values for the first recursion): The integral over the product of three associated Legendre polynomials (with orders matching as shown below) is a necessary ingredient when developing products of Legendre polynomials into a series linear in the Legendre polynomials. {\displaystyle P_{2}^{2}} {\displaystyle \geq } In that case the parameters are usually labelled with Greek letters. ≥ ) : In terms of θ, {\displaystyle \sin \theta } {\displaystyle [0,\pi ]} The definitions of Pℓ±m, resulting from this expression by substitution of ±m, are proportional. 0 h cos μ In general, when ℓ and m are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are not polynomials when m is odd. ( ℓ ) 1 �oKc�{����]�ޯv}d�u>r��b�p�N�a����(,���3���tH������F&Ȁ�ԥ�����f�Р�(�p)�l��2�D�H��~�u�6̩��pKA'>��mS��p����P3�)7n sin where the indices ℓ and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. the angle − }[tpֳ��oڧғ>��c3����O�=�0�W"�qs���] �-��W�y���R� >��2�����6�~�)]��㷋4 ��v�f� �^�y��W_�y�PzXY���J(�T�W� 9��ped̾"�Ʌ���t��8YV��� 4�k��&�b,8��d��A7:�l#X��qf'�Sf��#��=(X�\wu�?=�],8��@���t�[ Bs�n�\$�Y�%Xx�5�i6}�����O����#Ƣ���SE9!�r��~Hd������axBU*7�������nL�^��ղ�lh}�ok}�I�C%�>�d%KX�/��p��u�� ��:փR9x�*Аs�}��Q;Y�u윒i�q~n� 5-&Se %AF�s�;!��Q T�"@# B�>C*1���"+��c�%�Zc�ٍ�Y�Jr�ͦ �W'Zr���!������ҟƅ[2?ƭ��`���܀�D ��Bv�O ��@�ĩ �-Վ�����rJ.G[���(R'�0 In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation (−) − + [(+) − −] =,or equivalently [(−) ()] + [(+) − −] =,where the indices ℓ and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. m ) x nonsingular solutions only when P nIѣ"I˴GZ=�R��O|�' pr�!�%�p��ub��]���2��������a�F� AT�"�k+�|8�?���tsr�. stream What makes these functions useful is that they are central to the solution of the equation is the hypergeometric function. Q , appears in a multiplying factor. = 2 P P {\displaystyle P_{\lambda }^{\mu }(z)} + For example, ��|c λ ��� [email protected]� ℓ 1 /Filter /FlateDecode 1 cos {\displaystyle \cos(m\phi )} ) ) = ℓ λ θ For instance, this turns out to be necessary when doing atomic calculations of the Hartree–Fock variety where matrix elements of the Coulomb operator are needed. Together, they make a set of functions called spherical harmonics. θ Workers in the fields of geodesy, geomagnetism and spectral analysis use a different phase and normalization factor than given here (see spherical harmonics).

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