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He showed that if the asymptotic series for the integral as $x\\rightarrow+\\infty$ is truncated after $rx$ terms, where $0<r<R$, then the resulting remainder is exponentially small of order $O(e^{-rx})$. In this note we extend this result to include situations when $f(t)$ has a branch point at $t=0$ and when $x$ is a complex variabl"},"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":"1505.06905","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-05-26T11:27:02Z","cross_cats_sorted":[],"title_canon_sha256":"789170c24d440691d297d1b9e691291bca3500be6abd515ec6f8aa6a8f7ed0d7","abstract_canon_sha256":"ec21c2d676d61ce88f5c3d689793ac406de6a1760bf47d7c365ca17fcf99652c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:27.606305Z","signature_b64":"xEZhuSXl2SqwL7Hg+qdzkrEkHbaYhtPs54u8slclNnq+Ieo9P+qxwdbsDnFjlKYSSkpvO5ulN7A38+0H3nHOCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6e04331217558704a63ddf6b27886427a1df79a6843e810ca23c62b9545b5fd","last_reissued_at":"2026-05-18T02:03:27.605708Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:27.605708Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On an extension of Watson's lemma due to Ursell","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"R. 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