A Gaussian quadrature-like formula for numerical estimation of integrals. It uses weighting
function 
in the interval ![[-1,1]](/images/equations/ChebyshevQuadrature/Inline2.svg)
and forces all the weights to be equal. The general formula
is
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(1)
|
where the abscissas 
are found by taking terms up to 
in the Maclaurin series
of
![s_n(y)=exp{1/2n[-2+ln(1-y)(1-1/y)+ln(1+y)(1+1/y)]},](/images/equations/ChebyshevQuadrature/NumberedEquation2.svg) |
(2)
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and then defining
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(3)
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The roots of 
then give the abscissas.
The first few values are
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(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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(13)
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Because the roots are all real for 
and 
only (Hildebrand 1956), these are the only permissible orders for Chebyshev quadrature.
The error term is
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(14)
|
where
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(15)
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The first few values of 
are 2/3, 8/45, 1/15, 32/945, 13/756, and 16/1575 (Hildebrand
1956). Beyer (1987) gives abscissas up to 
and Hildebrand (1956) up to 
.
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| 2 |  |
| 3 | 0 |
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| 4 |  |
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| 5 | 0 |
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| 6 |  |
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| 7 | 0 |
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| 9 | 0 |
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The abscissas and weights can be computed analytically for small 
.
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| 2 |  |
| 3 | 0 |
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| 4 |  |
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| 5 | 0 |
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