Pith Number

pith:DPZZESCS

pith:2026:DPZZESCSUNRX4YRLIU6XUY22X7

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A Generalized Closed Form of Ramanujan-Type Fourier Cosine Transform via Meijer's G-Function

R. P. Paris, S. A. Dar

The Ramanujan integral R_C(m,n) equals an infinite series of Meijer G-functions under convergence conditions on m and n.

arxiv:2605.13882 v1 · 2026-05-11 · math.NT

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Claims

C1strongest claim

We obtain analytical evaluations of the Ramanujan integral R_C(m,n) subject to suitable convergence conditions in terms of an infinite series of Meijer G-functions of one variable, by using Mellin-Barnes-type contour integral representations of the cosine function.

C2weakest assumption

The interchange of integration order and the absolute convergence of the resulting contour integrals under the stated conditions on m and n.

C3one line summary

The Ramanujan integral R_C(m,n) equals an infinite series of Meijer G-functions, with generalizations and closed forms for nine related series.

References

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[1] C., and Straub, A 2021

[2] Berndt, B. C. ; Integrals associated with Ramanujan and e lliptic functions. The Ramanujan Journal, 1, 2016 2016

[3] Carslaw, H. S. ; Introduction to the theory of F ourier’s series and integral s. Macmillan and co., limited st. Martin’s street, London, 1921 1921

[4] Ditkin, V .A. and Prudnikov, A.P . ; Integral transforms and operational calculus . Pergamon Press, Oxford, London, Frankfurt, 1965 1965

[5] and Paris, R.B.; On integrals involving quotie nts of hyperbolic functions, Journal of the Ramanujan Mathematical Society, 36(1), 1-10, 2021 2021

Formal links

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Receipt and verification

First computed	2026-05-17T23:39:19.172614Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

1bf3924852a3637e622b453d7a635abffe79a5ee110df1bbbcf37f472e5fb8e5

Aliases

arxiv: 2605.13882 · arxiv_version: 2605.13882v1 · doi: 10.48550/arxiv.2605.13882 · pith_short_12: DPZZESCSUNRX · pith_short_16: DPZZESCSUNRX4YRL · pith_short_8: DPZZESCS

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/DPZZESCSUNRX4YRLIU6XUY22X7 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 1bf3924852a3637e622b453d7a635abffe79a5ee110df1bbbcf37f472e5fb8e5

Canonical record JSON

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    "abstract_canon_sha256": "034669c2e92015ef3a271268a185ae77845cb88678799e32677e895a01ab69f4",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-11T08:10:36Z",
    "title_canon_sha256": "518c5e41b22b27f6b899201f7b03db2e077c2cd1135ed0977efb3ba069cb2475"
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  "source": {
    "id": "2605.13882",
    "kind": "arxiv",
    "version": 1
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