Interpolation for numerical integration

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The discussion centers on finding an effective interpolation method for the function Γ(t), which is known only at discrete points, to facilitate exact integration of the integral involving tan(Γ(t)t + φ). Simpson's rule is suggested as a strong candidate among numerical integration methods. The user seeks an interpolation that allows for analytical integration, potentially using approximations like Taylor or sine series. The goal is to derive an expression that can be integrated analytically rather than numerically. The conversation emphasizes the need for a suitable interpolation technique to achieve this.
kaniello
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Hallo,
\Gamma(t) is a function that i can know only at dicrete points and appears in this integral:
\int\Gamma(t) * tan (\Gamma(t)*t + \varphi)
My question is now, which could be the best interpolation of \Gamma(t) that would allopw an exact integration?
Thank you very much in advance
 
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Simpon's rule is definitely a good option. What I was looking for is the following: imagine that in the place of \Gamma(t) you have an approximation, like a Taylor series, or sine series or whatever, that would allow an analytical integration of this approximated expression.
 

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