{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Note that the result of each integration is " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "multiplied" } ] }, { "__type": "Text", "__tag": 4046, "value": " by " }, { "__type": "ParamRef", "__tag": 4071, "name": "scl" }, { "__type": "Text", "__tag": 4046, "value": ". Why is this important to note? Say one is making a linear change of variable " }, { "__type": "InlineMath", "__tag": 4057, "value": "u = ax + b" }, { "__type": "Text", "__tag": 4046, "value": " in an integral relative to " }, { "__type": "InlineRole", "__tag": 4003, "value": "x", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ". Then " }, { "__type": "InlineMath", "__tag": 4057, "value": "dx = du/a" }, { "__type": "Text", "__tag": 4046, "value": ", so one will need to set " }, { "__type": "ParamRef", "__tag": 4071, "name": "scl" }, { "__type": "Text", "__tag": 4046, "value": " equal to " }, { "__type": "InlineMath", "__tag": 4057, "value": "1/a" }, { "__type": "Text", "__tag": 4046, "value": " - perhaps not what one would have first thought." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Also note that, in general, the result of integrating a C-series needs to be \"reprojected\" onto the C-series basis set. Thus, typically, the result of this function is \"unintuitive,\" albeit correct; see Examples section below." } ] } ], "title": [], "level": 0, "target": null }, "Warns": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Raises": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "", "annotation": "ValueError", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "InlineCode", "__tag": 4051, "value": "m < 0" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineCode", "__tag": 4051, "value": "len(k) > m" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineCode", "__tag": 4051, "value": "np.ndim(lbnd) != 0" }, { "__type": "Text", "__tag": 4046, "value": ", or " }, { "__type": "InlineCode", "__tag": 4051, "value": "np.ndim(scl) != 0" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Yields": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Methods": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Returns": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "S", "annotation": "ndarray", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Hermite_e series coefficients of the integral." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Integrate a Hermite_e series." } ] } ], "title": [], "level": 0, "target": null }, "Receives": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Warnings": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Attributes": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Parameters": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "c", "annotation": "array_like", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Array of Hermite_e series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "m", "annotation": "int, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Order of integration, must be positive. (Default: 1)" } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "k", "annotation": "{[], list, scalar}, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Integration constant(s). The value of the first integral at " }, { "__type": "InlineCode", "__tag": 4051, "value": "lbnd" }, { "__type": "Text", "__tag": 4046, "value": " is the first value in the list, the value of the second integral at " }, { "__type": "InlineCode", "__tag": 4051, "value": "lbnd" }, { "__type": "Text", "__tag": 4046, "value": " is the second value, etc. If " }, { "__type": "InlineCode", "__tag": 4051, "value": "k == []" }, { "__type": "Text", "__tag": 4046, "value": " (the default), all constants are set to zero. If " }, { "__type": "InlineCode", "__tag": 4051, "value": "m == 1" }, { "__type": "Text", "__tag": 4046, "value": ", a single scalar can be given instead of a list." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "lbnd", "annotation": "scalar, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The lower bound of the integral. (Default: 0)" } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "scl", "annotation": "scalar, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Following each integration the result is " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "multiplied" } ] }, { "__type": "Text", "__tag": 4046, "value": " by " }, { "__type": "ParamRef", "__tag": 4071, "name": "scl" }, { "__type": "Text", "__tag": 4046, "value": " before the integration constant is added. (Default: 1)" } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "axis", "annotation": "int, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Axis over which the integral is taken. (Default: 0)." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Returns the Hermite_e series coefficients " }, { "__type": "ParamRef", "__tag": 4071, "name": "c" }, { "__type": "Text", "__tag": 4046, "value": " integrated " }, { "__type": "ParamRef", "__tag": 4071, "name": "m" }, { "__type": "Text", "__tag": 4046, "value": " times from " }, { "__type": "ParamRef", "__tag": 4071, "name": "lbnd" }, { "__type": "Text", "__tag": 4046, "value": " along " }, { "__type": "ParamRef", "__tag": 4071, "name": "axis" }, { "__type": "Text", "__tag": 4046, "value": ". At each iteration the resulting series is " }, { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "multiplied" } ] }, { "__type": "Text", "__tag": 4046, "value": " by " }, { "__type": "ParamRef", "__tag": 4071, "name": "scl" }, { "__type": "Text", "__tag": 4046, "value": " and an integration constant, " }, { "__type": "ParamRef", "__tag": 4071, "name": "k" }, { "__type": "Text", "__tag": 4046, "value": ", is added. The scaling factor is for use in a linear change of variable. (\"Buyer beware\": note that, depending on what one is doing, one may want " }, { "__type": "ParamRef", "__tag": 4071, "name": "scl" }, { "__type": "Text", "__tag": 4046, "value": " to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument " }, { "__type": "ParamRef", "__tag": 4071, "name": "c" }, { "__type": "Text", "__tag": 4046, "value": " is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series " }, { "__type": "InlineCode", "__tag": 4051, "value": "H_0 + 2*H_1 + 3*H_2" }, { "__type": "Text", "__tag": 4046, "value": " while [[1,2],[1,2]] represents " }, { "__type": "InlineCode", "__tag": 4051, "value": "1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)" }, { "__type": "Text", "__tag": 4046, "value": " if axis=0 is " }, { "__type": "InlineCode", "__tag": 4051, "value": "x" }, { "__type": "Text", "__tag": 4046, "value": " and axis=1 is " }, { "__type": "InlineCode", "__tag": 4051, "value": "y" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ], "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": "build-install/usr/lib/python3.14/site-packages/numpy/polynomial/hermite_e.py", "item_line": 675, "item_type": "function", "aliases": [ "numpy.polynomial.HermiteE._int", "numpy.polynomial.hermite_e.hermeint" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Code", "__tag": 4050, "value": "from numpy.polynomial.hermite_e import hermeint\nhermeint([1, 2, 3]) # integrate once, value 0 at 0.\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "hermeint([1, 2, 3], m=2) # integrate twice, value & deriv 0 at 0\n", "execution_status": "failure" }, { "__type": "Code", "__tag": 4050, "value": "hermeint([1, 2, 3], k=1) # integrate once, value 1 at 0.\nhermeint([1, 2, 3], lbnd=-1) # integrate once, value 0 at -1\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "hermeint([1, 2, 3], m=2, k=[1, 2], lbnd=-1)\n", "execution_status": "failure" } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "hermeder", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "numpy.polynomial.hermite_e:hermeder" }, "kind": "module" }, "descriptions": [], "type": null } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "c", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "m", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "1" }, { "__type": "SigParam", "__tag": 4030, "name": "k", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "[]" }, { "__type": "SigParam", "__tag": 4030, "name": "lbnd", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "0" }, { "__type": "SigParam", "__tag": 4030, "name": "scl", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "1" }, { "__type": "SigParam", "__tag": 4030, "name": "axis", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "0" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "hermeint" }, "references": null, "qa": "numpy.polynomial.hermite_e:hermeint", "arbitrary": [], "local_refs": [ "S", "axis", "c", "k", "lbnd", "m", "scl" ] }