{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:UTXHQQZRPX3FCDMQUJKMMWANQW","short_pith_number":"pith:UTXHQQZR","schema_version":"1.0","canonical_sha256":"a4ee7843317df6510d90a254c6580d859180807f07ddf843b64d67eeb59ba6f7","source":{"kind":"arxiv","id":"1508.04376","version":1},"attestation_state":"computed","paper":{"title":"Saddle-point integration of $C_\\infty$ \"bump\" functions","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Steven G. Johnson","submitted_at":"2015-08-18T16:44:21Z","abstract_excerpt":"This technical note describes the application of saddle-point integration to the asymptotic Fourier analysis of the well-known $C_\\infty$ \"bump\" function $\\exp[-(1-x^2)^{-1}]$, deriving both the asymptotic decay rate $k^{-3/4} \\exp(-\\sqrt k)$ of the Fourier transform $F(k)$ and the exact coefficient. The result is checked against brute-force numerical integration and is extended to generalizations of this bump function."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1508.04376","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CV","submitted_at":"2015-08-18T16:44:21Z","cross_cats_sorted":[],"title_canon_sha256":"135e4706ce651d41d3c8fd23e19d58195fbb6ee5d7e1ce7d2470afcd58c3573f","abstract_canon_sha256":"1f1909ad985fe5db7a5e0b81e8dbda73d46bbad766c9819cf4bb08947e0f7711"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:05.499135Z","signature_b64":"f0hCnHunWQtKQOcbp7qr5hAtVB84lT2F5D8u/mMvB50/nUIBUu3WK8zhmsw8y36QsfXSu7JaS1I+GkjiGe5TDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a4ee7843317df6510d90a254c6580d859180807f07ddf843b64d67eeb59ba6f7","last_reissued_at":"2026-05-18T01:35:05.498557Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:05.498557Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Saddle-point integration of $C_\\infty$ \"bump\" functions","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Steven G. Johnson","submitted_at":"2015-08-18T16:44:21Z","abstract_excerpt":"This technical note describes the application of saddle-point integration to the asymptotic Fourier analysis of the well-known $C_\\infty$ \"bump\" function $\\exp[-(1-x^2)^{-1}]$, deriving both the asymptotic decay rate $k^{-3/4} \\exp(-\\sqrt k)$ of the Fourier transform $F(k)$ and the exact coefficient. The result is checked against brute-force numerical integration and is extended to generalizations of this bump function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.04376","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1508.04376","created_at":"2026-05-18T01:35:05.498649+00:00"},{"alias_kind":"arxiv_version","alias_value":"1508.04376v1","created_at":"2026-05-18T01:35:05.498649+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.04376","created_at":"2026-05-18T01:35:05.498649+00:00"},{"alias_kind":"pith_short_12","alias_value":"UTXHQQZRPX3F","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"UTXHQQZRPX3FCDMQ","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"UTXHQQZR","created_at":"2026-05-18T12:29:44.643036+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UTXHQQZRPX3FCDMQUJKMMWANQW","json":"https://pith.science/pith/UTXHQQZRPX3FCDMQUJKMMWANQW.json","graph_json":"https://pith.science/api/pith-number/UTXHQQZRPX3FCDMQUJKMMWANQW/graph.json","events_json":"https://pith.science/api/pith-number/UTXHQQZRPX3FCDMQUJKMMWANQW/events.json","paper":"https://pith.science/paper/UTXHQQZR"},"agent_actions":{"view_html":"https://pith.science/pith/UTXHQQZRPX3FCDMQUJKMMWANQW","download_json":"https://pith.science/pith/UTXHQQZRPX3FCDMQUJKMMWANQW.json","view_paper":"https://pith.science/paper/UTXHQQZR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1508.04376&json=true","fetch_graph":"https://pith.science/api/pith-number/UTXHQQZRPX3FCDMQUJKMMWANQW/graph.json","fetch_events":"https://pith.science/api/pith-number/UTXHQQZRPX3FCDMQUJKMMWANQW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UTXHQQZRPX3FCDMQUJKMMWANQW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UTXHQQZRPX3FCDMQUJKMMWANQW/action/storage_attestation","attest_author":"https://pith.science/pith/UTXHQQZRPX3FCDMQUJKMMWANQW/action/author_attestation","sign_citation":"https://pith.science/pith/UTXHQQZRPX3FCDMQUJKMMWANQW/action/citation_signature","submit_replication":"https://pith.science/pith/UTXHQQZRPX3FCDMQUJKMMWANQW/action/replication_record"}},"created_at":"2026-05-18T01:35:05.498649+00:00","updated_at":"2026-05-18T01:35:05.498649+00:00"}