Editing Trapezoidal rule (section)

=== Periodic and peak functions ===
The trapezoidal rule converges rapidly for periodic functions. This is an easy consequence of the [[Euler-Maclaurin summation formula]], which says that
if <math>f</math> is <math>p</math> times continuously differentiable with period <math>T</math>
<math display="block">\sum_{k=0}^{N-1} f(kh)h =
    \int_0^T f(x)\,dx  +
    \sum_{k=1}^{\lfloor p/2\rfloor} \frac{B_{2k}}{(2k)!} (f^{(2k - 1)}(T) - f^{(2k - 1)}(0)) - (-1)^p h^p \int_0^T\tilde{B}_{p}(x/T)f^{(p)}(x) \, dx
</math>
where <math>h:=T/N</math> and <math>\tilde{B}_{p}</math> is the periodic extension of the <math>p</math>th Bernoulli polynomial.<ref>{{cite book|title=Numerical Analysis, volume 181 of Graduate Texts in Mathematics|first=Rainer|last=Kress |year=1998 |publisher=Springer-Verlag}}</ref> Due to the periodicity, the derivatives at the endpoint cancel and we see that the error is <math>O(h^p)</math>.

A similar effect is available for peak-like functions, such as [[Gaussian function|Gaussian]], [[Exponentially modified Gaussian distribution|Exponentially modified Gaussian]] and other functions with derivatives at integration limits that can be neglected.<ref>{{Cite journal |last=Goodwin|first=E. T. |date=1949 |title=The evaluation of integrals of the form <math>\textstyle\int_{-\infty}^\infty{f(x)e^{-x^2}dx}</math> | journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |language=en |volume=45 |issue=2 |pages=241–245 |doi=10.1017/S0305004100024786 |bibcode=1949PCPS...45..241G |issn=1469-8064}}</ref> The evaluation of the full integral of a Gaussian function by trapezoidal rule with 1% accuracy can be made using just 4 points.<ref name=":0">{{Cite journal| last1=Kalambet|first1=Yuri |last2=Kozmin|first2=Yuri |last3=Samokhin|first3=Andrey |date=2018 |title=Comparison of integration rules in the case of very narrow chromatographic peaks |journal=Chemometrics and Intelligent Laboratory Systems|volume=179 |pages=22–30 |doi=10.1016/j.chemolab.2018.06.001|issn=0169-7439}}</ref> [[Simpson's rule]] requires 1.8 times more points to achieve the same accuracy.<ref name=":0" /><ref name="w02" />