{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:VE6QTPHWZDUQJ3WMFGN5OVMZD3","short_pith_number":"pith:VE6QTPHW","schema_version":"1.0","canonical_sha256":"a93d09bcf6c8e904eecc299bd755991ecb90070108960e568d77943a31370fac","source":{"kind":"arxiv","id":"2204.02729","version":2},"attestation_state":"computed","paper":{"title":"A contribution to the mathematical theory of diffraction. Part I: A note on double Fourier integrals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CV","math.MP"],"primary_cat":"math.AP","authors_text":"Andrey I. Korolkov, Andrey V. Shanin, Rapha\\\"el C. Assier","submitted_at":"2022-04-06T11:10:33Z","abstract_excerpt":"We consider a large class of physical fields $u$ written as double inverse Fourier transforms of some functions $F$ of two complex variables. Such integrals occur very often in practice, especially in diffraction theory. Our aim is to provide a closed-form far-field asymptotic expansion of $u$. In order to do so, we need to generalise the well-established complex analysis notion of contour indentation to integrals of functions of two complex variables. It is done by introducing the so-called bridge and arrow notation. Thanks to another integration surface deformation, we show that, to achieve "},"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":"2204.02729","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2022-04-06T11:10:33Z","cross_cats_sorted":["math-ph","math.CV","math.MP"],"title_canon_sha256":"fc623ff45e4ac2751f472c4875783a663e3b9e5cbd1f406d6f58840402c7160d","abstract_canon_sha256":"2cb067d2c3aa991945aef77ceaf6047572b6dd3852e8ae010ad095a4039e4def"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:06:56.377051Z","signature_b64":"frOlAM76d4ldl4eScrJ869Yl8Z6/gPl59uM4NLcXcbfHyR8UZhRtqa7XlhNzm7r1a7xf1waeWVhF2ufOHh8bBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a93d09bcf6c8e904eecc299bd755991ecb90070108960e568d77943a31370fac","last_reissued_at":"2026-07-05T05:06:56.376652Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:06:56.376652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A contribution to the mathematical theory of diffraction. Part I: A note on double Fourier integrals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CV","math.MP"],"primary_cat":"math.AP","authors_text":"Andrey I. Korolkov, Andrey V. Shanin, Rapha\\\"el C. Assier","submitted_at":"2022-04-06T11:10:33Z","abstract_excerpt":"We consider a large class of physical fields $u$ written as double inverse Fourier transforms of some functions $F$ of two complex variables. Such integrals occur very often in practice, especially in diffraction theory. Our aim is to provide a closed-form far-field asymptotic expansion of $u$. In order to do so, we need to generalise the well-established complex analysis notion of contour indentation to integrals of functions of two complex variables. It is done by introducing the so-called bridge and arrow notation. Thanks to another integration surface deformation, we show that, to achieve "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2204.02729","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2204.02729/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2204.02729","created_at":"2026-07-05T05:06:56.376709+00:00"},{"alias_kind":"arxiv_version","alias_value":"2204.02729v2","created_at":"2026-07-05T05:06:56.376709+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2204.02729","created_at":"2026-07-05T05:06:56.376709+00:00"},{"alias_kind":"pith_short_12","alias_value":"VE6QTPHWZDUQ","created_at":"2026-07-05T05:06:56.376709+00:00"},{"alias_kind":"pith_short_16","alias_value":"VE6QTPHWZDUQJ3WM","created_at":"2026-07-05T05:06:56.376709+00:00"},{"alias_kind":"pith_short_8","alias_value":"VE6QTPHW","created_at":"2026-07-05T05:06:56.376709+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.28271","citing_title":"Which Saddles Contribute? The South-East Rule for Multidimensional Integrals","ref_index":48,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VE6QTPHWZDUQJ3WMFGN5OVMZD3","json":"https://pith.science/pith/VE6QTPHWZDUQJ3WMFGN5OVMZD3.json","graph_json":"https://pith.science/api/pith-number/VE6QTPHWZDUQJ3WMFGN5OVMZD3/graph.json","events_json":"https://pith.science/api/pith-number/VE6QTPHWZDUQJ3WMFGN5OVMZD3/events.json","paper":"https://pith.science/paper/VE6QTPHW"},"agent_actions":{"view_html":"https://pith.science/pith/VE6QTPHWZDUQJ3WMFGN5OVMZD3","download_json":"https://pith.science/pith/VE6QTPHWZDUQJ3WMFGN5OVMZD3.json","view_paper":"https://pith.science/paper/VE6QTPHW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2204.02729&json=true","fetch_graph":"https://pith.science/api/pith-number/VE6QTPHWZDUQJ3WMFGN5OVMZD3/graph.json","fetch_events":"https://pith.science/api/pith-number/VE6QTPHWZDUQJ3WMFGN5OVMZD3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VE6QTPHWZDUQJ3WMFGN5OVMZD3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VE6QTPHWZDUQJ3WMFGN5OVMZD3/action/storage_attestation","attest_author":"https://pith.science/pith/VE6QTPHWZDUQJ3WMFGN5OVMZD3/action/author_attestation","sign_citation":"https://pith.science/pith/VE6QTPHWZDUQJ3WMFGN5OVMZD3/action/citation_signature","submit_replication":"https://pith.science/pith/VE6QTPHWZDUQJ3WMFGN5OVMZD3/action/replication_record"}},"created_at":"2026-07-05T05:06:56.376709+00:00","updated_at":"2026-07-05T05:06:56.376709+00:00"}