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Default is " }, { "__type": "CrossRef", "__tag": 4002, "value": "False", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "builtins", "version": "?", "kind": "module", "path": "builtins:False" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Defined as " }, { "__type": "InlineMath", "__tag": 4057, "value": "P^*_n(x) = P_n(2x - 1)" }, { "__type": "Text", "__tag": 4046, "value": " for " }, { "__type": "InlineMath", "__tag": 4057, "value": "P_n" }, { "__type": "Text", "__tag": 4046, "value": " the nth Legendre polynomial." } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "/scipy/special/_orthogonal.py", "item_line": 2761, "item_type": "function", "aliases": [ "scipy.special.sh_legendre" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "The shifted Legendre polynomials :math:`P_n^*` are related\nto the non-shifted polynomials :math:`P_n` by\n:math:`P_n^*(x) = P_n(2x - 1)`. We can verify this on the\ninterval :math:`[0, 1]`:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy.special import sh_legendre, legendre\nfrom scipy.integrate import trapezoid\nx = np.arange(0.0, 1.0, 0.01)\nn = 3\nnp.allclose(sh_legendre(n)(x), legendre(n)(2*x - 1))\n", "execution_status": "success", "out": "True\n" }, { "__type": "Text", "__tag": 4046, "value": "\nThe polynomials :math:`P_n^*` satisfy a recurrence\nrelation obtained by the change of variables\n:math:`t = 2x - 1` in the standard Legendre recurrence:\n\n.. math::\n\n (n+1) P_{n+1}^*(x) = (2n+1)(2x-1)\\,P_n^*(x) - n\\,P_{n-1}^*(x).\n\nThis can be easily checked on :math:`[0, 1]`\nfor :math:`n = 3`:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "n = 3\nx = np.linspace(0.0, 1.0, 101)\nlhs = (n + 1) * sh_legendre(n + 1)(x)\nrhs = (\n (2*n + 1) * (2*x - 1) * sh_legendre(n)(x)\n - n * sh_legendre(n - 1)(x)\n)\nnp.allclose(lhs, rhs)\n", "execution_status": "success", "out": "True\n" }, { "__type": "Text", "__tag": 4046, "value": "\nOrthogonality over :math:`[0,1]` with weight 1 can be\nchecked numerically; for example, :math:`P_2^*`\nis orthogonal to :math:`P_3^*`:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "x = np.linspace(0.0, 1.0, 400)\ny = sh_legendre(2)(x) * sh_legendre(3)(x)\n", "execution_status": "success", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "np.isclose(trapezoid(y, x), 0.0, atol=1e-12)\n", "execution_status": "failure", "out": "np.True_\n" } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "scipy.special.legendre", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.special._orthogonal:legendre" }, "kind": "module" }, "descriptions": [], "type": null }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "scipy.special.roots_sh_legendre", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "scipy.special.roots_sh_legendre" }, "kind": "module" }, "descriptions": [], "type": null } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "n", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "monic", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "False" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "sh_legendre" }, "references": null, "qa": "scipy.special._orthogonal:sh_legendre", "arbitrary": [], "local_refs": [ "P", "monic", "n" ] }