{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This function computes a discrete version of the Hankel transform" } ] }, { "__type": "Math", "__tag": 4058, "value": "a(r) = \\int_{0}^{\\infty} \\! A(k) \\, J_\\mu(kr) \\, r \\, dk \\;," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "J_\\mu" }, { "__type": "Text", "__tag": 4046, "value": " is the Bessel function of order " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\mu" }, { "__type": "Text", "__tag": 4046, "value": ". The index " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\mu" }, { "__type": "Text", "__tag": 4046, "value": " may be any real number, positive or negative. 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To calculate the evaluation\npoints in ascending order, flip the order of ``k`` in the equation\nfor ``r``.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "r = 1/np.flip(k)\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nCompare ``a`` with the kernel function corresponding to ``k[240]``.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "a_f = jv(2, k[240]*r)*r\nfig, ax = plt.subplots()\n", "execution_status": "success", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "ax.plot(r, a, label=\"ifht\")\nax.plot(r, a_f, \"--\", label=r\"$r J_2(k_{240} r)$\")\nax.set_xlabel(r\"$r$\")\nax.set_ylabel(\"amplitude\")\nax.legend()\n", "execution_status": "failure", "out": "\n" }, { "__type": "Code", "__tag": 4050, "value": "plt.show()\n", "execution_status": "success", "out": "" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "2.0.0.dev0+git20260907.41eeb59", "kind": "assets", "path": "fig-61e6dd7d526f5f6e.png" } }, { "__type": "Text", "__tag": 4046, "value": "\nThe plot shows that, for larger values of ``r``, the inverse Hankel transform\nfollows the kernel function closely." } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "fht", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.fft._fftlog:fht" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Definition of the fast Hankel transform." } ] } ], "type": "func" }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "fhtoffset", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.fft._fftlog_backend:fhtoffset" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Return an optimal offset for " }, { "__type": "InlineRole", "__tag": 4003, "value": "ifht", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ], "type": "func" } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "A", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "dln", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "mu", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "offset", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "0.0" }, { "__type": "SigParam", "__tag": 4030, "name": "bias", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "0.0" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "ifht" }, "references": null, "qa": "scipy.fft._fftlog:ifht", "arbitrary": [], "local_refs": [ "A", "a", "bias", "dln", "mu", "offset" ] }