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\title{Filon's Integration Formula}

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                        Filon's Integration Formula[3]

25.4.47[1]

$$ \int_{x_0}^{x_{2n}} f(x) \cos tx \, dx = h \biggl[
   \alpha(th) (f_{2n} \sin t x_{2n} - f_{0} \sin t x_0) +
   \beta(th) C_{2n} +
   \gamma(th) C_{2n-1} +
   {2 \over 45}th^4 S'_{2n-1}\biggr]-R_n $$

25.4.48

$$ C_{2n} = \sum_{i=0}^{n} f_{2i} \cos (tx_{2i})-{1 \over 2}
[f_{2n} \cos tx_{2n} + f_0 \cos t x_0] $$

25.4.49

$$ C_{2n-1} = \sum_{i=1}^{n} f_{2i-1} \cos t x_{2i-1} $$

25.4.50

$$ S'_{2n-1} = \sum_{i=1}^{n} f_{2i-1}^{(3)} \sin tx_{2i-1} $$

25.4.51

$$ R_n = {1 \over 90}nh^5 f^{(4)}(\xi) + O(th^7) $$

25.4.52

$$\eqnarray{
 \alpha(\theta) &= {1 \over \theta} + {\sin 2\theta \over 2 \theta^2} -
                   {2 \sin^2 \theta \over \theta^3}\cr
 \beta(\theta)  &= 2 \biggl( {1 + \cos^2 \theta \over \theta^2} -
                   {\sin 2 \theta \over \theta^3}\biggr)\cr
 \gamma(\theta) &= 4 \biggl( {\sin \theta \over \theta^3} -
                   {\cos \theta \over \theta^2}\biggr)\cr} 
}$$

For small $\theta$ we have

25.4.53

$$\eqnarray{
 \alpha &= {2\theta^3 \over 45} - {2\theta^5 \over 315} +
           {2\theta^7 \over 4725} - \cdots \cr
 \beta  &= {2 \over 3} + {2\theta^2 \over 15} - {4\theta^4 \over 105} +
           {2\theta^6 \over 567} - \cdots \cr
 \gamma &= {4 \over 3} - {2\theta^2 \over 15} + {\theta^4 \over 210} -
           {\theta^6 \over 11340} + \cdots \cr}
}$$

25.4.54

$$ \int_{x_0}^{x_{2n}} f(x) \sin tx \, dx = h \biggl[
\alpha(th) (f_0 \cos tx_0 - f_{2n} \cos tx_{2n}) +
\beta S_{2n} + \gamma S_{2n-1} + {2 \over 45} th^4 C'_{2n-1}\biggr] - R_n $$

25.4.55

$$ S_{2n} = \sum_{i=0}^{n} f_{2i} \sin (tx_{2i}) -
{1 \over 2} [f_{2n} \sin(tx_{2n})+f_0\sin(tx_0)] $$

25.4.56

$$ S_{2n-1} = \sum_{i=1}^{n} f_{2i-1} \sin (tx_{2i-1}) $$

25.4.57

$$ C'_{2n-1} = \sum_{i=1}^{n} f_{2i-1}^{(3)} \cos(tx_{2i-1}) $$

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