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<p>A theorem of Biermann and Weierstrass addresses the differential equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((w')^2 = f(w)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>w</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in which <i>f</i> is a quartic polynomial whose zeros are simple, expressing its solutions in terms of the Weierstrass <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\wp \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>℘</mi> </math></EquationSource> </InlineEquation>-function whose invariants are those of the quartic. We offer an elementary proof of this theorem, based on ideas of Goursat.</p>

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A theorem of Biermann and Weierstrass

  • 期刊论文

  • 发表日期：

    2026年4月27日

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  • P. L. Robinson

摘要

A theorem of Biermann and Weierstrass addresses the differential equation \((w')^2 = f(w)\) ( w ′ ) 2 = f ( w ) in which f is a quartic polynomial whose zeros are simple, expressing its solutions in terms of the Weierstrass \(\wp \) ℘ -function whose invariants are those of the quartic. We offer an elementary proof of this theorem, based on ideas of Goursat.