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Mehrem","submitted_at":"2011-10-27T18:11:46Z","abstract_excerpt":"The integrals $\\threej{\\Llo}{\\Llt}{\\Llth} {0}{0}{0}\\,\\int_0^\\infty \\,r^{\\Llth+1}\\,e^{-\\alpha r}\\,j_\\Llo(k_1r)\\, j_\\Llt(k_2r)\\,dr$ and $\\threej{\\Llo}{\\Llt}{\\Llth} {0}{0}{0}\\,\\int_0^\\infty \\,r^{\\Llth+2}\\,e^{-\\alpha r}\\,j_\\Llo(k_1r)\\, j_\\Llt(k_2r)\\,dr$ are evaluated analytically. The result is a finite sum over the associated Legendre function of the second kind."},"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":"1110.6147","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-10-27T18:11:46Z","cross_cats_sorted":["math.MP","nucl-th"],"title_canon_sha256":"48bc9d3dec5df55d8028296273e11961a3fdf187b54a7a515b2a60030e25ab29","abstract_canon_sha256":"f8830a71e935768d2ee27c544c8bbfcbb3014bede87167f955ec26b36b67e3c2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:09.459362Z","signature_b64":"+vMwinMOq379nhoGb8sRduObqei8vIHPhorLkx5qTOUE2vrv/aF/T+oaAQkqbNH+zA9bouWgYOXhJ7sD82J+CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"153375c1edd7c35e4bc9903288945ea257c43e7d1db11231c81f15a86904c23b","last_reissued_at":"2026-05-18T04:10:09.458666Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:09.458666Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An Analytical Evaluation For The Integral Of Two Spherical Bessel Functions With An Additional Exponential And Polynomial Factor","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","nucl-th"],"primary_cat":"math-ph","authors_text":"R. Mehrem","submitted_at":"2011-10-27T18:11:46Z","abstract_excerpt":"The integrals $\\threej{\\Llo}{\\Llt}{\\Llth} {0}{0}{0}\\,\\int_0^\\infty \\,r^{\\Llth+1}\\,e^{-\\alpha r}\\,j_\\Llo(k_1r)\\, j_\\Llt(k_2r)\\,dr$ and $\\threej{\\Llo}{\\Llt}{\\Llth} {0}{0}{0}\\,\\int_0^\\infty \\,r^{\\Llth+2}\\,e^{-\\alpha r}\\,j_\\Llo(k_1r)\\, j_\\Llt(k_2r)\\,dr$ are evaluated analytically. The result is a finite sum over the associated Legendre function of the second kind."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6147","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":"1110.6147","created_at":"2026-05-18T04:10:09.458758+00:00"},{"alias_kind":"arxiv_version","alias_value":"1110.6147v1","created_at":"2026-05-18T04:10:09.458758+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.6147","created_at":"2026-05-18T04:10:09.458758+00:00"},{"alias_kind":"pith_short_12","alias_value":"CUZXLQPN27BV","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_16","alias_value":"CUZXLQPN27BV4S6J","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_8","alias_value":"CUZXLQPN","created_at":"2026-05-18T12:26:26.731475+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/CUZXLQPN27BV4S6JSAZIRFC6UJ","json":"https://pith.science/pith/CUZXLQPN27BV4S6JSAZIRFC6UJ.json","graph_json":"https://pith.science/api/pith-number/CUZXLQPN27BV4S6JSAZIRFC6UJ/graph.json","events_json":"https://pith.science/api/pith-number/CUZXLQPN27BV4S6JSAZIRFC6UJ/events.json","paper":"https://pith.science/paper/CUZXLQPN"},"agent_actions":{"view_html":"https://pith.science/pith/CUZXLQPN27BV4S6JSAZIRFC6UJ","download_json":"https://pith.science/pith/CUZXLQPN27BV4S6JSAZIRFC6UJ.json","view_paper":"https://pith.science/paper/CUZXLQPN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1110.6147&json=true","fetch_graph":"https://pith.science/api/pith-number/CUZXLQPN27BV4S6JSAZIRFC6UJ/graph.json","fetch_events":"https://pith.science/api/pith-number/CUZXLQPN27BV4S6JSAZIRFC6UJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CUZXLQPN27BV4S6JSAZIRFC6UJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CUZXLQPN27BV4S6JSAZIRFC6UJ/action/storage_attestation","attest_author":"https://pith.science/pith/CUZXLQPN27BV4S6JSAZIRFC6UJ/action/author_attestation","sign_citation":"https://pith.science/pith/CUZXLQPN27BV4S6JSAZIRFC6UJ/action/citation_signature","submit_replication":"https://pith.science/pith/CUZXLQPN27BV4S6JSAZIRFC6UJ/action/replication_record"}},"created_at":"2026-05-18T04:10:09.458758+00:00","updated_at":"2026-05-18T04:10:09.458758+00:00"}