{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:EN2YFG44YGL3C7Z4J7LJBY5YR7","short_pith_number":"pith:EN2YFG44","schema_version":"1.0","canonical_sha256":"2375829b9cc197b17f3c4fd690e3b88ff84391a77a32aa8de43b3d664bb00e63","source":{"kind":"arxiv","id":"math/0604026","version":1},"attestation_state":"computed","paper":{"title":"Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity","license":"","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"D.Karp, S.M.Sitnik","submitted_at":"2006-04-03T09:57:45Z","abstract_excerpt":"We find two convergent series expansions for Legendre's first incomplete elliptic integral $F(\\lambda,k)$ in terms of recursively computed elementary functions. Both expansions are valid at every point of the unit square $0<\\lambda,k<1$. Truncated expansions yield asymptotic approximations for $F(\\lambda,k)$ as $\\lambda$ and/or $k$ tend to unity, including the case when logarithmic singularity $\\lambda=k=1$ is approached from any direction. Explicit error bounds are given at every order of approximation. For the reader's convenience we present explicit expressions for low-order approximations "},"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":"math/0604026","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"2006-04-03T09:57:45Z","cross_cats_sorted":[],"title_canon_sha256":"5c5dbeea8852a81e1bd67ddecf0bcd16de083d458e80dc04232bd144a0f1a150","abstract_canon_sha256":"d720f7a54ce851c97b43312f8932987f3dd399ae75d9cad835e885ceb4a67830"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:25.042079Z","signature_b64":"9XdA9xTTu0AsN4/iHODVIHhYUGmfqlAOZhV6MoA7sm+M2j7C3T8DGyrf+epQ6ed/mbeGZrUUzs6jBYvLMs2wDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2375829b9cc197b17f3c4fd690e3b88ff84391a77a32aa8de43b3d664bb00e63","last_reissued_at":"2026-05-18T01:04:25.041595Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:25.041595Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity","license":"","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"D.Karp, S.M.Sitnik","submitted_at":"2006-04-03T09:57:45Z","abstract_excerpt":"We find two convergent series expansions for Legendre's first incomplete elliptic integral $F(\\lambda,k)$ in terms of recursively computed elementary functions. Both expansions are valid at every point of the unit square $0<\\lambda,k<1$. Truncated expansions yield asymptotic approximations for $F(\\lambda,k)$ as $\\lambda$ and/or $k$ tend to unity, including the case when logarithmic singularity $\\lambda=k=1$ is approached from any direction. Explicit error bounds are given at every order of approximation. For the reader's convenience we present explicit expressions for low-order approximations "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0604026","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":"math/0604026","created_at":"2026-05-18T01:04:25.041672+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0604026v1","created_at":"2026-05-18T01:04:25.041672+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0604026","created_at":"2026-05-18T01:04:25.041672+00:00"},{"alias_kind":"pith_short_12","alias_value":"EN2YFG44YGL3","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_16","alias_value":"EN2YFG44YGL3C7Z4","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_8","alias_value":"EN2YFG44","created_at":"2026-05-18T12:25:53.939244+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/EN2YFG44YGL3C7Z4J7LJBY5YR7","json":"https://pith.science/pith/EN2YFG44YGL3C7Z4J7LJBY5YR7.json","graph_json":"https://pith.science/api/pith-number/EN2YFG44YGL3C7Z4J7LJBY5YR7/graph.json","events_json":"https://pith.science/api/pith-number/EN2YFG44YGL3C7Z4J7LJBY5YR7/events.json","paper":"https://pith.science/paper/EN2YFG44"},"agent_actions":{"view_html":"https://pith.science/pith/EN2YFG44YGL3C7Z4J7LJBY5YR7","download_json":"https://pith.science/pith/EN2YFG44YGL3C7Z4J7LJBY5YR7.json","view_paper":"https://pith.science/paper/EN2YFG44","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0604026&json=true","fetch_graph":"https://pith.science/api/pith-number/EN2YFG44YGL3C7Z4J7LJBY5YR7/graph.json","fetch_events":"https://pith.science/api/pith-number/EN2YFG44YGL3C7Z4J7LJBY5YR7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EN2YFG44YGL3C7Z4J7LJBY5YR7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EN2YFG44YGL3C7Z4J7LJBY5YR7/action/storage_attestation","attest_author":"https://pith.science/pith/EN2YFG44YGL3C7Z4J7LJBY5YR7/action/author_attestation","sign_citation":"https://pith.science/pith/EN2YFG44YGL3C7Z4J7LJBY5YR7/action/citation_signature","submit_replication":"https://pith.science/pith/EN2YFG44YGL3C7Z4J7LJBY5YR7/action/replication_record"}},"created_at":"2026-05-18T01:04:25.041672+00:00","updated_at":"2026-05-18T01:04:25.041672+00:00"}