Abstract
If EquationSource<m:math display='block'> <m:mrow> <m:mi>P</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>τ</m:mi> <m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:msub> <m:mi>a</m:mi> <m:mn>0</m:mn> </m:msub> <m:msup> <m:mi>τ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>+</m:mo><m:msub> <m:mi>a</m:mi> <m:mn>1</m:mn> </m:msub> <m:msup> <m:mi>τ</m:mi> <m:mrow> <m:mi>n</m:mi><m:mo>−</m:mo><m:mn>1</m:mn></m:mrow> </m:msup> <m:mo>+</m:mo><m:msub> <m:mi>a</m:mi> <m:mn>2</m:mn> </m:msub> <m:msup> <m:mi>τ</m:mi> <m:mrow> <m:mi>n</m:mi><m:mo>−</m:mo><m:mn>2</m:mn></m:mrow> </m:msup> <m:mn>...</m:mn><m:mo>+</m:mo><m:msub> <m:mi>a</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>=</m:mo><m:msub> <m:mi>a</m:mi> <m:mn>0</m:mn> </m:msub> <m:mrow><m:mo>(</m:mo> <m:mrow> <m:mi>τ</m:mi><m:mo>−</m:mo><m:msub> <m:mi>r</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> <m:mo>)</m:mo></m:mrow><m:mrow><m:mo>(</m:mo> <m:mrow> <m:mi>τ</m:mi><m:mo>−</m:mo><m:msub> <m:mi>r</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> <m:mo>)</m:mo></m:mrow><m:mn>...</m:mn><m:mrow><m:mo>(</m:mo> <m:mrow> <m:mi>τ</m:mi><m:mo>−</m:mo><m:msub> <m:mi>r</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> <m:mo>)</m:mo></m:mrow></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$P\left( \tau \right) = {a_0}{\tau ^n} + {a_1}{\tau ^{n - 1}} + {a_2}{\tau ^{n - 2}}... + {a_n} = {a_0}\left( {\tau - {r_1}} \right)\left( {\tau - {r_2}} \right)...\left( {\tau - {r_n}} \right)$$ andEquationSource<m:math display='block'> <m:mrow> <m:msub> <m:mi>R</m:mi> <m:mn>1</m:mn> </m:msub> <m:mrow><m:mo>[</m:mo> <m:mrow> <m:mi>τ</m:mi><m:mo>,</m:mo><m:msqrt> <m:mrow> <m:mi>P</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>τ</m:mi> <m:mo>)</m:mo></m:mrow></m:mrow> </m:msqrt> </m:mrow> <m:mo>]</m:mo></m:mrow></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${R_1}\left[ {\tau ,\sqrt {P\left( \tau \right)} } \right]$$ is a rational function of τ and of EquationSource<m:math display='block'> <m:mrow> <m:msqrt> <m:mrow> <m:mi>P</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>τ</m:mi> <m:mo>)</m:mo></m:mrow></m:mrow> </m:msqrt> <m:mo>,</m:mo></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\sqrt {P\left( \tau \right)} ,$$ ,the general integral1
is called a hyperelliptic integral when n is greater than four. If the degree of P(τ) is equal to 2p+ 2, one can always obtain by means of a rational transformation (e.g., τ =r 1 +1/t) an equivalent integral in which the radicand is of degree 2p +1.
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See H. Lenz: Zurückführung einiger Integrale auf einfachere. Sitzungsberichte der Bayerischen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse, No. 10, 1951, pp. 73 - 80.
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© 1954 Springer-Verlag Berlin Heidelberg
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Byrd, P.F., Friedman, M.D. (1954). Hyperelliptic Integrals. In: Handbook of Elliptic Integrals for Engineers and Physicists. Die Grundlehren der Mathematischen Wissenschaften, vol 67. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-52803-3_10
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DOI: https://doi.org/10.1007/978-3-642-52803-3_10
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