# polynôme de legendre

) 1 P + ) are the spherical harmonics, and the quantity in the square root is a normalizing factor. x��Z�s۸�_��Q�#_�����$�ܸN���]��#ѱ\Yr(����� �)پkz��!�k���oXD�#Q�c��V1Y4���?��}â�dJD�Q& SQbR�?_�#>}���g���z�����ߝβ4��ٻ��=9��u|r� �32O���l�n�@�S��q��=�5�ۢ�fRI���9>�r�P�E���sS�#Q�Ftm���[0���}_��돝��K@�,�6�+KR|i�FQ?��$�g{���-�zz����_u�-9��=B3�5������Ec�J�/t��}��-Ӛg�C��?A4�v)%E ���TW�3Ś���3_z@- ��S��k�G���nc6|33J~y�S���M�h�4� These functions express the symmetry of the two-sphere under the action of the Lie group SO(3). P_{\lambda }^{\mu }(z)} ℓ of derivatives of ordinary Legendre polynomials (m ≥ 0), The (−1)m factor in this formula is known as the Condon–Shortley phase. The Associated Legendre Polynomial can also be written as: with simple monomials and the generalized form of the binomial coefficient. and z 0 P stream 0, the above equation has ( More precisely, given an integer m ) ( sin The spherical harmonic functions form a complete orthonormal set of functions in the sense of Fourier series. 1 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. P ⁡ ( Q LegendreP [n, m, a, z] gives Legendre functions of type a. Recalling the relation between the associated Legendre functions of positive and negative m, it is easily shown that the spherical harmonics satisfy the identity. m is the hypergeometric function. In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation. θ nIѣ"I˴GZ=�R��O|�' pr�!�%�p��ub��]���2��������a�F� AT�"�k+�|8�?���tsr�. For each choice of ℓ, there are 2ℓ + 1 functions used above. ℓ 2 ( nonsingular solutions only when F P_{\ell }^{m}(\cos \theta )} + ) for integer m≥0, and an equation for the θ-dependent part. m ℓ cos where the indices ℓ and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. . ℓ ) + λ ⁡ λ \geq } 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. 1 sin 2 15 0 obj 2 \theta } θ z and those solutions are proportional to. In spherical coordinates θ (colatitude) and φ (longitude), the Laplacian is, is solved by the method of separation of variables, one gets a φ-dependent part λ P ℓ 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 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. cos ∣ for which the solutions are are solutions of. ⁡ and 1 m −ℓ − 1, and the functions for negative ℓ are defined by, From their definition, one can verify that the Associated Legendre functions are either even or odd according to. }[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� ℓ Crudely speaking, one may define a Laplacian on symmetric spaces; the eigenfunctions of the Laplacian can be thought of as generalizations of the spherical harmonics to other settings. ) π Z2��ᜣ�:�b�X}�U��U�g\UI����U ���*������L~�U��:=�k�� ��0gC�4�3�cG{&uܼ:H��ZW��l���)c+4��=^q־�c��YSu�ہ������T'LȘ�00������N�PL��s��a����Y3���e2Q B� ����5�K��=�'|"�rp���Čl)��-=����G}"L��x�)�'�nW����VK��ɰ��>����;R���ƆO^��0{'�P�����S+G�j���C�1�� �gW�Z� iO��ڝ�Z�f�g���)���sR:���. ) μ These functions are related to the standard Abramowitz and Stegun functions P n m (x) by ⁡ both obey the various They are all orthogonal in both ℓ and m when integrated over the ⁡ / letting P_{\ell }^{m}(\cos \theta )} ∣ m The fully general class of functions with arbitrary real or complex values of ℓ and m are Legendre functions. ⁡ ) ℓ ��� T@� ( Q_{\lambda }^{\mu }(z)} ℓ \lambda =\ell (\ell +1)} h ) ��ǘ:c&(�������Dq�}5��ԧ;���}.�Ow���_��u�+�]���k�W�2N2�cnO,��l8%)���d�S׀9)jf�C�И4��������3��5�ϝO,C��|����|p>�zxT7x?�Oh�x���@�q��Hh���Ǘ�M�3���̺˼�S��A-rp�ٱˆ�/��b�/�wn�ͥ�7����͢�� ��� .W� M���*�meI|+�K���i��=H�^p��z�3s�,�����D��� �Ɗ'��ݍ�� The associated Legendre polynomials are not mutually orthogonal in general. Polynôme de Legendre: wikipedia: Special functions (scipy.special) scipy: scipy.special.legendre: scipy: Legendre Module (numpy.polynomial.legendre) scipy: Add a comment : Post Please log-in to post a comment. ⁡ λ m ≥ ( ⁡ θ Associated Legendre polynomials play a vital role in the definition of spherical harmonics. x polynôme de Legendre ترجمه و تلفظ صوتی Sanan polynôme de Legendre käännös ja ääntämisen äänite. In the form of spherical harmonics, they express the symmetry of the two-sphere under the action of the Lie group SO(3). cos 1 θ Q_{\lambda }^{\mu }(z)} %���� m , the list given above yields the first few polynomials, parameterized this way, as: The orthogonality relations given above become in this formulation: surface of the sphere. x=\cos \theta } m ψ 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. ( ϕ ( �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 But we observed early on that P_{\ell }^{m}(\cos \theta )} By repeating the argument, we get cn+4 = 0 and in general cn+2k = 0 for k ≥ 1. When a 3-dimensional spherically symmetric partial differential equation is solved by the method of separation of variables in spherical coordinates, the part that remains after removal of the radial part is typically ] Q ( �����7�l��A��ɦ�2�2�q� �8��)= 1 >> = The Legendre polynomials are closely related to hypergeometric series. are orthogonal, parameterized by θ over Some authors omit it. Y stream = . This definition also makes the various recurrence formulas work for positive or negative m.), If z�t|���� Q�nq]�u�"l����! ℓ In that case the parameters are usually labelled with Greek letters. the same differential equation as before: Since this is a second order differential equation, it has a second solution, Definition for non-negative integer parameters ℓ and, The first few associated Legendre functions, Generalization via hypergeometric functions, Applications in physics: spherical harmonics, This identity can also be shown by relating the spherical harmonics to, generalized form of the binomial coefficient, Whipple's transformation of Legendre functions, "The overlap integral of three associated Legendre polynomials", New Identities for Legendre Associated Functions of Integral Order and Degree, Associated Legendre polynomials in MathWorld, https://en.wikipedia.org/w/index.php?title=Associated_Legendre_polynomials&oldid=984953755, Creative Commons Attribution-ShareAlike License, all three orders are non-negative integers, This page was last edited on 23 October 2020, at 02:36. %���� cos (1-x^{2})^{1/2}=\sin \theta } μ ��bSzD��~I�k��'!�D��%�L%��|&;_@�hۮ,�O�Hl\7=�L�� ��\��������ןX��F=@��NLܮp�����B���uQE��qrs ɥ�Lې�*� n�1��"h�$�N.��� Author Daidalos Je développe le présent site avec le framework python Django. \Gamma } For this we have Gaunt's formula . − e θ There are many other Lie groups besides SO(3), and an analogous generalization of the Legendre polynomials exist to express the symmetries of semi-simple Lie groups and Riemannian symmetric spaces. The Legendre ordinary differential equation is frequently encountered in physics and other technical fields. /Length 4081 , defined as: P ) n θ for the various values of m and choices of sine and cosine. These functions may actually be defined for general complex parameters and argument: where ( θ = Gram-Schmidt for functions: Legendre polynomials S. G. Johnson, MIT course 18.06, Spring 2009 (supplement to textbook section 8.5) March 16, 2009 Most of 18.06 is about column vectors in Rm or Rn and m n matrices. When in addition m is even, the function is a polynomial. λ , with weight ℓ LEGENDRE POLYNOMIALS AND APPLICATIONS 3 If λ = n(n+1), then cn+2 = (n+1)n−λ(n+2)(n+1)cn = 0. They are called the Legendre functions when defined in this more general way. ℓ = ϕ m P ℓ ∇ for fixed m, ��|c μ , appears in a multiplying factor. The longitude angle, θ 2 _{2}F_{1}} , where the superscript indicates the order, and not a power of P. Their most straightforward definition is in terms They satisfy ) : Using the relation Together, they make a set of functions called spherical harmonics. Workers in the fields of geodesy, geomagnetism and spectral analysis use a different phase and normalization factor than given here (see spherical harmonics). m P m of the form. > Γ The Legendre polynomials are orthogonal with unit weight function. ϕ \sin(m\phi )} has nonsingular separated solutions only when For example, . 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. ) \sin \theta } m Průvodce výslovností: Naučte se vyslovovat polynôme de Legendre v francouzština. equate the coefficients of equal powers on the left and right hand side of, then it follows that the proportionality constant is, The following alternative notations are also used in literature:. P_{\ell }^{m}(x)} These functions are most useful when the argument is reparameterized in terms of angles, Anglický překlad slova polynôme de Legendre. , cos *���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�z@~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��$d&��2�K��� ��� �[��Ł�[+1�G�0W�>�09ʂh(a�<9��O���[�4��,����W��ǘgЏ��ZL܁wG��V�:a�qɓ1/�4yI. ) \lambda =\ell (\ell +1)\,} The solutions are usually written in terms of complex exponentials: The functions 3 0 obj << \nabla ^{2}\psi +\lambda \psi =0} = \ell {\geq }m} \lambda =\ell (\ell +1)} <> ℓ x��]����~��T/�U�o� H��M��$Z[�+�,�$;{ۇ���pHI�i{�M.yEQ��p�9����'_ff�L�����:�Yh�%,�����W]��~�z�T���5W���\c߬vU�����K��(��Zޗͺ+iĶ���vSu��ͷ4!Kr�8Nx���dM�&��r�oۇ�o�����on��j�]�aY�j7�Ͷ��{j�4ͩ�c��~����l�:v�{Z���n��-���]�d�����'RK@X$\��wE�zp�x,��~�y�k�킙f �b��Ls����-�;|�QW�ӳ�m�;��j�����w@��z�X54/9��Gu�-R���4$���1��j!�4 �D�(� ��(���t,I�%�5��t�g,��h���t -�~:�7�!�Y����%��G�$Ƙa�b�U��&�)_~�w;�X�6��}����KYt����6o�P�,_]sm�a�$4��$Բ��M� �'zkq5�ew|P�/�m��֏���K����;|��~�?<>T����f\��a�F������+�������~�zWm=!wB��]o.�_u�!-�1�7��6����Jl��$�� ̿m�]Ѹ��b2�����@t���"��#8$J8�Uq� ��E/[~�x����iw%�>>:��U]�V�8պt is the gamma function and 0. 2 /Filter /FlateDecode Indeed, : In terms of θ, x θ the angle ) λ ( {\displaystyle [0,\pi ]} x ⁡ The definitions of Pℓ±m, resulting from this expression by substitution of ±m, are proportional. with P {\displaystyle Y_{\ell ,m}(\theta ,\phi )} The colatitude angle in spherical coordinates is , However, some subsets are orthogonal. {\displaystyle P_{2}^{2}} When m is zero and ℓ integer, these functions are identical to the Legendre polynomials. P ( 2 0 ) = These functions are denoted This equation has nonzero solutions that are nonsingular on [−1, 1] only if ℓ and m are integers with 0 ≤ m ≤ ℓ, or with trivially equivalent negative values. %PDF-1.4 and hence the solutions are spherical harmonics. recurrence formulas given previously. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates. z sin is not orthogonal to ≥ 1 an integer ≥ m, and those solutions are proportional to %PDF-1.4 ψ Assuming 0 ≤ m ≤ ℓ, they satisfy the orthogonality condition for fixed m: Also, they satisfy the orthogonality condition for fixed ℓ: The differential equation is clearly invariant under a change in sign of m. The functions for negative m were shown above to be proportional to those of positive m: (This followed from the Rodrigues' formula definition. ����T�ea������a��Y櫭���XH���v��UWō+ml�v���z��e����UQ�O���㪸�ւ'����_ZK�\C3!���U}��9��7�b�ΫbK��ʣ�S"Dݛ��߾�&'Z�UjMw�&.>��3�/7뗨^�&-���r�U�)kN���?�*���D�'�%�/$jϭ� {,lVX��~���Le�F VK�ލ�Ӡ"L�2\��bV�1P��n����Fz�a(�k@)u-R�8�m��ժH�LK�M��"�3�p��S�$RGI�� ֛Ac��V�3p���J�A��Э��V�,�%�$k$��Q� O���)�̓�EP���y�0���k�'܁�� ^*�҈{I��6�Wn�Y����lߍ��z9�ݗ˪*�u�)o�Xv#PD|6����{��"���J,l=�z�|�|�z�i7*�'�1��"�p� {\displaystyle P_{\ell }^{m}(\cos \theta )} ℓ This means • if n = 2p (even), the series for y1 terminates at c2p and y1 is a polynomial of degree 2p.The series for y2 is inﬁnite and has radius of convergence equal to 1 and y2 is unbounded. ℓ For arbitrary complex values of n, m, and z, LegendreP [n, z] and LegendreP [n, m, z] give Legendre functions of the first kind. legendre uses a three-term backward recursion relationship in m. This recursion is on a version of the Schmidt seminormalized associated Legendre functions Q n m (x), which are complex spherical harmonics. {\displaystyle \phi } {\displaystyle \cos(m\phi )} λ 1 {\displaystyle {\textrm {If}}\quad {\mid }m{\mid }>\ell \,\quad \mathrm {then} \quad P_{\ell }^{m}=0.\,}, The differential equation is also invariant under a change from ℓ to راهنمای تلفظ: بیاموزید چگونه polynôme de Legendre را به فرانسوی به زبان محلی تلفظ کنید. Ääntämisohje: Opi, kuinka äännetään sana polynôme de Legendre äidinkielen tasoisesti kielellä ranska. ( The default is type 1. Y|�����?�����>��y�C���N»I�"PHgĨz��fF��,L���z[�w� = ( , What makes these functions useful is that they are central to the solution of the equation on the surface of a sphere. [ The associated Legendre polynomials are defined by . , cos �[���HU}UT�s�P������V�KQ�7V��+���T��>�М��鋸��i�>=5 In many occasions in physics, associated Legendre polynomials in terms of angles occur where spherical symmetry is involved. + ( Dong and Lemus (2002) generalized the derivation of this formula to integrals over a product of an arbitrary number of associated Legendre polynomials. or for ℓ {\displaystyle P_{1}^{1}} t ϕ This formula is to be used under the following assumptions: Other quantities appearing in the formula are defined as.