The Area Under the Witch of Agnesi

The area under the witch of Agnesi is discussed. Real-variable proofs of typically appeal to the fact that the inverse tangent is an anti-derivative for the integrand or that the differential equation tan = 1 + tan2 holds on the open interval. Complex-variable proofs typically appeal to Cauchy'...

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Veröffentlicht in:	The American mathematical monthly 2024-10, Vol.131 (9), p.802-802
1. Verfasser:	Bradley, David M.
Format:	Artikel
Sprache:	eng
Schlagworte:	Derivatives Differential equations Integrals Mathematical problems Proof theory Theorems Transcendental functions
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description	The area under the witch of Agnesi is discussed. Real-variable proofs of typically appeal to the fact that the inverse tangent is an anti-derivative for the integrand or that the differential equation tan = 1 + tan2 holds on the open interval. Complex-variable proofs typically appeal to Cauchy's residue theorem. The proof below requires neither Cauchy's theorem nor knowledge of any transcendental functions, and also connects the value of the integral more directly with the geometrical definition of pi.
doi_str_mv	10.1080/00029890.2024.2386921
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subjects	Derivatives Differential equations Integrals Mathematical problems Proof theory Theorems Transcendental functions
title	The Area Under the Witch of Agnesi
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