n H F = − ∑ r = 1 n 1 α r d θ r d t 1 θ r {\displaystyle {\frac {n\mathrm {H} }{\mathrm {F} }}=-\sum _{\mathrm {r} =1}^{\mathrm {n} }{\frac {1}{\alpha _{\mathrm {r} }}}{\frac {d\theta _{\mathrm {r} }}{dt}}{\frac {1}{\theta }}_{\mathrm {r} }{}} ,
quindi la (5) diviene
∑ 1 μ 1 α r ∫ f ( x ) ( x − a ) R ( x ) n d x = − Π f ( x ) ( x − a ) R ( x ) n ∑ r = 1 n 1 α r log θ r ( x ) {\displaystyle \sum _{1}^{\mu }{\frac {1}{\alpha _{\mathrm {r} }}}\int {\frac {f(x)}{(x-a){\sqrt[{\mathrm {n} }]{\mathrm {R} (x)}}}}dx=-\Pi {\frac {f(x)}{(x-a){\sqrt[{\mathrm {n} }]{\mathrm {R} (x)}}}}\sum _{\mathrm {r} =1}^{\mathrm {n} }{\frac {1}{\alpha _{\mathrm {r} }}}\log \theta _{\mathrm {r} }(x)} + f ( a ) R ( a ) n ∑ 1 n 1 α r log θ r ( a ) + C o s t . e . {\displaystyle +{\frac {f(a)}{\sqrt[{\mathrm {n} }]{\mathrm {R} (a)}}}\sum _{1}^{\mathrm {n} }{\frac {1}{\alpha _{\mathrm {r} }}}\log \theta _{\mathrm {r} }(a)+\mathrm {Cost.^{e}} .}
In questo risultato consiste appunto il teorema di Abel.
,
![{\displaystyle \sum _{1}^{\mu }{\frac {1}{\alpha _{\mathrm {r} }}}\int {\frac {f(x)}{(x-a){\sqrt[{\mathrm {n} }]{\mathrm {R} (x)}}}}dx=-\Pi {\frac {f(x)}{(x-a){\sqrt[{\mathrm {n} }]{\mathrm {R} (x)}}}}\sum _{\mathrm {r} =1}^{\mathrm {n} }{\frac {1}{\alpha _{\mathrm {r} }}}\log \theta _{\mathrm {r} }(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b691c74a923f30c7c58fc0ee6bdcf249ea0a278a)
![{\displaystyle +{\frac {f(a)}{\sqrt[{\mathrm {n} }]{\mathrm {R} (a)}}}\sum _{1}^{\mathrm {n} }{\frac {1}{\alpha _{\mathrm {r} }}}\log \theta _{\mathrm {r} }(a)+\mathrm {Cost.^{e}} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/058eaef99624372dff92ecaf209184ca0cb63ee1)
