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method</b> is a technique for <a href="/wiki/Numerical_integration" title="Numerical integration">numerical integration</a> of <a href="/wiki/Oscillation_(mathematics)" title="Oscillation (mathematics)">oscillatory</a> integrals. It is named after English mathematician <a href="/wiki/Louis_Napoleon_George_Filon" title="Louis Napoleon George Filon">Louis Napoleon George Filon</a>, who first described the method in 1934.<sup id="cite_ref-filon-1930_1-0" class="reference"><a href="#cite_note-filon-1930-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Filon_quadrature&action=edit&section=1" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The method is applied to oscillatory definite integrals in the form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)g(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)g(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a10c656f6ec1ed212b155c9c6750b0ae259806f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.007ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)g(x)dx}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a982c6635ab3b98d9e12d5f5a8533359bcb38a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\textstyle f(x)}"></span> is a relatively slowly-varying function and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b78b479cc1c1ccd06f8cdfd31223335921e5a5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\textstyle g(x)}"></span> is either <a href="/wiki/Sine_and_cosine" title="Sine and cosine">sine or cosine</a> or a complex exponential that causes the rapid oscillation of the integrand, particularly for high frequencies. In Filon quadrature, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a982c6635ab3b98d9e12d5f5a8533359bcb38a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\textstyle f(x)}"></span> is divided into <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc5e93da2deafa5044f89898c38684fc37c154f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.226ex; height:2.176ex;" alt="{\textstyle 2N}"></span> subintervals of length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13fda070627ca694f85f588a432f8158cc4df1e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\textstyle h}"></span>, which are then <a href="/wiki/Interpolation" title="Interpolation">interpolated</a> by <a href="/wiki/Parabola" title="Parabola">parabolas</a>. Since each subinterval is now converted into a <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier integral</a> of <a href="/wiki/Quadratic_function" title="Quadratic function">quadratic polynomials</a>, these can be evaluated in closed-form by <a href="/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a>. For the case of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle g(x)=\cos(kx)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle g(x)=\cos(kx)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14eb6893449c727e9ad2ddd55b12463bd4d642b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.815ex; height:2.843ex;" alt="{\textstyle g(x)=\cos(kx)}"></span>, the integration formula is given as:<sup id="cite_ref-filon-1930_1-1" class="reference"><a href="#cite_note-filon-1930-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-davis-rabinowitz-1984_2-0" class="reference"><a href="#cite_note-davis-rabinowitz-1984-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\cos(kx)dx\approx h(\alpha \left[f(b)\sin(kb)-f(a)\sin(ka)\right]+\beta C_{2n}+\gamma C_{2n-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>≈</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mrow> <mo>[</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mi>β</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mi>γ</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\cos(kx)dx\approx h(\alpha \left[f(b)\sin(kb)-f(a)\sin(ka)\right]+\beta C_{2n}+\gamma C_{2n-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9388586a3112cb87e5a8190a9987abcbdca81384" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:71.793ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\cos(kx)dx\approx h(\alpha \left[f(b)\sin(kb)-f(a)\sin(ka)\right]+\beta C_{2n}+\gamma C_{2n-1})}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\left(\theta ^{2}+\theta \sin(\theta )\cos(\theta )-2\sin ^{2}(\theta )\right)/\theta ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\left(\theta ^{2}+\theta \sin(\theta )\cos(\theta )-2\sin ^{2}(\theta )\right)/\theta ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd3d6c650efceee4650d9c81bbb3657584832f44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.226ex; height:3.343ex;" alt="{\displaystyle \alpha =\left(\theta ^{2}+\theta \sin(\theta )\cos(\theta )-2\sin ^{2}(\theta )\right)/\theta ^{3}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =2\left[\theta (1+\cos ^{2}(\theta ))-2\sin(\theta )\cos(\theta )\right]/\theta ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>[</mo> <mrow> <mi>θ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =2\left[\theta (1+\cos ^{2}(\theta ))-2\sin(\theta )\cos(\theta )\right]/\theta ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c67c761dfc286446ebf9091b1015d3aaa1902cc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.125ex; height:3.343ex;" alt="{\displaystyle \beta =2\left[\theta (1+\cos ^{2}(\theta ))-2\sin(\theta )\cos(\theta )\right]/\theta ^{3}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =4(\sin(\theta )-\theta \cos(\theta ))/\theta ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =4(\sin(\theta )-\theta \cos(\theta ))/\theta ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e075d1362a0eb6b313af6d358e4e5c004d44f82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.724ex; height:3.176ex;" alt="{\displaystyle \gamma =4(\sin(\theta )-\theta \cos(\theta ))/\theta ^{3}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{2n}={\frac {1}{2}}f(a)\cos(ka)+f(a+2h)\cos(k(a+2h))+f(a+4h)\cos(k(a+4h))+\ldots +{\frac {1}{2}}f(b)\cos(kb)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mi>h</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>4</mn> <mi>h</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>4</mn> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{2n}={\frac {1}{2}}f(a)\cos(ka)+f(a+2h)\cos(k(a+2h))+f(a+4h)\cos(k(a+4h))+\ldots +{\frac {1}{2}}f(b)\cos(kb)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4048ccfca396b0c1e5a1ffa5e256ac44f28fb6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:97.669ex; height:5.176ex;" alt="{\displaystyle C_{2n}={\frac {1}{2}}f(a)\cos(ka)+f(a+2h)\cos(k(a+2h))+f(a+4h)\cos(k(a+4h))+\ldots +{\frac {1}{2}}f(b)\cos(kb)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{2n-1}=f(a+h)\cos(k(a+h))+f(a+3h)\cos(k(a+3h))+\ldots +f(b-h)\cos(k(b-h))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>h</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{2n-1}=f(a+h)\cos(k(a+h))+f(a+3h)\cos(k(a+3h))+\ldots +f(b-h)\cos(k(b-h))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9b5d12378fa2b974a752e5242fb0729bd83b8d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:88.709ex; height:2.843ex;" alt="{\displaystyle C_{2n-1}=f(a+h)\cos(k(a+h))+f(a+3h)\cos(k(a+3h))+\ldots +f(b-h)\cos(k(b-h))}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =kh}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>k</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =kh}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/591b8acc17eb1e553ffcb826c38fbaca3f365bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.739ex; height:2.176ex;" alt="{\displaystyle \theta =kh}"></span></dd></dl> <p>Explicit Filon integration formulas for sine and complex exponential functions can be derived similarly.<sup id="cite_ref-davis-rabinowitz-1984_2-1" class="reference"><a href="#cite_note-davis-rabinowitz-1984-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The formulas above fail for small <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11744bd71a5eb6efe4f28e12ca57f874d82658c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\textstyle \theta }"></span> values due to <a href="/wiki/Catastrophic_cancellation" title="Catastrophic cancellation">catastrophic cancellation</a>;<sup id="cite_ref-chase-fosdick-1969_3-0" class="reference"><a href="#cite_note-chase-fosdick-1969-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> approximations must be in such cases to mitigate numerical errors, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta =1/6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \theta =1/6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fd22a842372245da2a59d13189ebaea635b7c9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.676ex; height:2.843ex;" alt="{\textstyle \theta =1/6}"></span> being recommended as a possible switchover point for 44-bit <a href="/wiki/Significand" title="Significand">mantissa</a>.<sup id="cite_ref-davis-rabinowitz-1984_2-2" class="reference"><a href="#cite_note-davis-rabinowitz-1984-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Modifications, extensions and generalizations of Filon quadrature have been reported in <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a> and <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a> literature; these are known as Filon-type integration methods.<sup id="cite_ref-iserles-norsett-2004_4-0" class="reference"><a href="#cite_note-iserles-norsett-2004-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-xiang-2007_5-0" class="reference"><a href="#cite_note-xiang-2007-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> These include Filon-<a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">trapezoidal</a><sup id="cite_ref-davis-rabinowitz-1984_2-3" class="reference"><a href="#cite_note-davis-rabinowitz-1984-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and Filon–<a href="/wiki/Clenshaw%E2%80%93Curtis_quadrature" title="Clenshaw–Curtis quadrature">Clenshaw–Curtis</a> methods.<sup id="cite_ref-dominguez-2011_6-0" class="reference"><a href="#cite_note-dominguez-2011-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Filon_quadrature&action=edit&section=2" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Filon quadrature is widely used in physics and engineering for robust computation of Fourier-type integrals. Applications include evaluation of oscillatory <a href="/wiki/Sommerfeld_identity" title="Sommerfeld identity">Sommerfeld integrals</a> for <a href="/wiki/Computational_electromagnetics" title="Computational electromagnetics">electromagnetic</a> and <a href="/wiki/Seismology" title="Seismology">seismic</a> problems in layered media<sup id="cite_ref-cerveny-ravi-1971_7-0" class="reference"><a href="#cite_note-cerveny-ravi-1971-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-mosig-gardiol-1983_8-0" class="reference"><a href="#cite_note-mosig-gardiol-1983-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-chew-1990_9-0" class="reference"><a href="#cite_note-chew-1990-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and numerical solution to steady <a href="/wiki/Incompressible_flow" title="Incompressible flow">incompressible flow</a> problems in <a href="/wiki/Fluid_mechanics" title="Fluid mechanics">fluid mechanics</a>,<sup id="cite_ref-dennis-chang-1970_10-0" class="reference"><a href="#cite_note-dennis-chang-1970-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> as well as various different problems in <a href="/wiki/Neutron_scattering" title="Neutron scattering">neutron scattering</a>,<sup id="cite_ref-grimley-1990_11-0" class="reference"><a href="#cite_note-grimley-1990-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a><sup id="cite_ref-fedotov-2023_12-0" class="reference"><a href="#cite_note-fedotov-2023-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Metallurgy" title="Metallurgy">metallurgy</a>.<sup id="cite_ref-thouless-1987_13-0" class="reference"><a href="#cite_note-thouless-1987-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Filon_quadrature&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Tanh-sinh_quadrature" title="Tanh-sinh quadrature">Tanh-sinh quadrature</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Filon_quadrature&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-filon-1930-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-filon-1930_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-filon-1930_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFFilon1930" class="citation journal cs1">Filon, L. N. G. (1930). "III.—On a Quadrature Formula for Trigonometric Integrals". <i>Proceedings of the Royal Society of Edinburgh</i>. <b>49</b>: 38–47. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0370164600026262">10.1017/S0370164600026262</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+Edinburgh&rft.atitle=III.%E2%80%94On+a+Quadrature+Formula+for+Trigonometric+Integrals&rft.volume=49&rft.pages=38-47&rft.date=1930&rft_id=info%3Adoi%2F10.1017%2FS0370164600026262&rft.aulast=Filon&rft.aufirst=L.+N.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFilon+quadrature" class="Z3988"></span></span> </li> <li id="cite_note-davis-rabinowitz-1984-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-davis-rabinowitz-1984_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-davis-rabinowitz-1984_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-davis-rabinowitz-1984_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-davis-rabinowitz-1984_2-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavisRabinowitz1984" class="citation book cs1"><a href="/wiki/Philip_J._Davis" title="Philip J. Davis">Davis, Philip J.</a>; <a href="/wiki/Philip_Rabinowitz_(mathematician)" title="Philip Rabinowitz (mathematician)">Rabinowitz, Philip</a> (1984). <i>Methods of Numerical Integration</i> (2 ed.). Academic Press. pp. 151–160. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781483264288" title="Special:BookSources/9781483264288"><bdi>9781483264288</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+Numerical+Integration&rft.pages=151-160&rft.edition=2&rft.pub=Academic+Press&rft.date=1984&rft.isbn=9781483264288&rft.aulast=Davis&rft.aufirst=Philip+J.&rft.au=Rabinowitz%2C+Philip&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFilon+quadrature" class="Z3988"></span></span> </li> <li id="cite_note-chase-fosdick-1969-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-chase-fosdick-1969_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChaseFosdick1969" class="citation journal cs1">Chase, Stephen M.; Fosdick, Lloyd D. (1969). "An algorithm for Filon quadrature". <i><a href="/wiki/Communications_of_the_ACM" title="Communications of the ACM">Communications of the ACM</a></i>. <b>12</b> (8): 453–457. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F363196.363209">10.1145/363196.363209</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=An+algorithm+for+Filon+quadrature&rft.volume=12&rft.issue=8&rft.pages=453-457&rft.date=1969&rft_id=info%3Adoi%2F10.1145%2F363196.363209&rft.aulast=Chase&rft.aufirst=Stephen+M.&rft.au=Fosdick%2C+Lloyd+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFilon+quadrature" class="Z3988"></span></span> </li> <li id="cite_note-iserles-norsett-2004-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-iserles-norsett-2004_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIserlesNørsett2004" class="citation journal cs1">Iserles, A.; Nørsett, S. 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"On quadrature methods for highly oscillatory integrals and their implementation". <i><a href="/wiki/BIT_Numerical_Mathematics" title="BIT Numerical Mathematics">BIT Numerical Mathematics</a></i>. <b>44</b> (4): 755–772. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10543-004-5243-3">10.1007/s10543-004-5243-3</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=BIT+Numerical+Mathematics&rft.atitle=On+quadrature+methods+for+highly+oscillatory+integrals+and+their+implementation&rft.volume=44&rft.issue=4&rft.pages=755-772&rft.date=2004&rft_id=info%3Adoi%2F10.1007%2Fs10543-004-5243-3&rft.aulast=Iserles&rft.aufirst=A.&rft.au=N%C3%B8rsett%2C+S.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFilon+quadrature" class="Z3988"></span></span> </li> <li id="cite_note-xiang-2007-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-xiang-2007_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFXiang2007" class="citation journal cs1">Xiang, Shuhuang (2007). "Efficient Filon-type methods for". <i><a href="/wiki/Numerische_Mathematik" title="Numerische Mathematik">Numerische Mathematik</a></i>. <b>105</b>: 633–658. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00211-006-0051-0">10.1007/s00211-006-0051-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Numerische+Mathematik&rft.atitle=Efficient+Filon-type+methods+for&rft.volume=105&rft.pages=633-658&rft.date=2007&rft_id=info%3Adoi%2F10.1007%2Fs00211-006-0051-0&rft.aulast=Xiang&rft.aufirst=Shuhuang&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFilon+quadrature" class="Z3988"></span></span> </li> <li id="cite_note-dominguez-2011-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-dominguez-2011_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDomínguezGrahamSmyshlyaev2011" class="citation journal cs1">Domínguez, V.; Graham, I. G.; Smyshlyaev, V. P. (2011). "Stability and error estimates for Filon–Clenshaw–Curtis rules for highly oscillatory integrals". <i><a href="/wiki/IMA_Journal_of_Numerical_Analysis" title="IMA Journal of Numerical Analysis">IMA Journal of Numerical Analysis</a></i>. <b>31</b> (4): 1253–1280. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fimanum%2Fdrq036">10.1093/imanum/drq036</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IMA+Journal+of+Numerical+Analysis&rft.atitle=Stability+and+error+estimates+for+Filon%E2%80%93Clenshaw%E2%80%93Curtis+rules+for+highly+oscillatory+integrals&rft.volume=31&rft.issue=4&rft.pages=1253-1280&rft.date=2011&rft_id=info%3Adoi%2F10.1093%2Fimanum%2Fdrq036&rft.aulast=Dom%C3%ADnguez&rft.aufirst=V.&rft.au=Graham%2C+I.+G.&rft.au=Smyshlyaev%2C+V.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFilon+quadrature" class="Z3988"></span></span> </li> <li id="cite_note-cerveny-ravi-1971-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-cerveny-ravi-1971_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFČervenýRavindra1971" class="citation book cs1">Červený, Vlastislav; Ravindra, Ravi (1971). <i>Theory of Seismic Head Waves</i>. 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(1983). "Analytical and numerical techniques in the Green's function treatment of microstrip antennas and scatterers". <i><a href="/wiki/Proceedings_of_the_Institution_of_Electrical_Engineers" title="Proceedings of the Institution of Electrical Engineers">IEE Proceedings H</a></i>. <b>130</b> (2): 175–182. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1049%2Fip-h-1.1983.0029">10.1049/ip-h-1.1983.0029</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEE+Proceedings+H&rft.atitle=Analytical+and+numerical+techniques+in+the+Green%27s+function+treatment+of+microstrip+antennas+and+scatterers&rft.volume=130&rft.issue=2&rft.pages=175-182&rft.date=1983&rft_id=info%3Adoi%2F10.1049%2Fip-h-1.1983.0029&rft.aulast=Mosig&rft.aufirst=J.+R.&rft.au=Gardiol%2C+F.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFilon+quadrature" class="Z3988"></span></span> </li> <li id="cite_note-chew-1990-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-chew-1990_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChew1990" class="citation book cs1"><a href="/wiki/Weng_Cho_Chew" title="Weng Cho Chew">Chew, Weng Cho</a> (1990). <i>Waves and Fields in Inhomogeneous Media</i>. 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