1 - ((sin pi x)/(pi x))^2

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$\scriptstyle 1-\mathrm {sinc} _{\pi }^{2}\,x\quad (-3\,\leq \,x\,\leq \,+3,\,\Delta x\,=\,0.05)\,$

1     $\scriptstyle 1-\mathrm {sinc} _{\pi }^{2}\,x\,$
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0

The elementary transcendental function

$1-\mathrm {sinc} _{\pi }^{2}\,x={\begin{cases}0&{\text{if }}x=0,\\1-\left({\frac {\sin \pi x}{\pi x}}\right)^{2}&{\text{otherwise}},\end{cases}}$

where

$\mathrm {sinc} _{\pi }\,x\equiv {\begin{cases}1&{\text{if }}x=0,\\{\frac {\sin \pi x}{\pi x}}&{\text{otherwise}},\end{cases}}\,$

is the normalized sinc function.

$\scriptstyle 1-\mathrm {sinc} _{\pi }^{2}\,x\,$ is the main part of the correlation function in Montgomery's pair correlation conjecture.

Maclaurin series expansion of 1 - ((sin pi x)/(pi x))^2

The Maclaurin series expansion of 1 - ((sin pi x)/(pi x))^2 is

$1-\left({\frac {\sin \pi x}{\pi x}}\right)^{2}={\frac {\pi ^{2}x^{2}}{3}}-{\frac {2\pi ^{4}x^{4}}{45}}+{\frac {\pi ^{6}x^{6}}{315}}-{\frac {2\pi ^{8}x^{8}}{14175}}+{\frac {2\pi ^{10}x^{10}}{467775}}-{\frac {4\pi ^{12}x^{12}}{42567525}}+{\frac {\pi ^{14}x^{14}}{638512875}}-{\frac {2\pi ^{16}x^{16}}{97692469875}}+{\frac {2\pi ^{18}x^{18}}{9280784638125}}-{\frac {4\pi ^{20}x^{20}}{2143861251406875}}+\cdots \,$

1 - ((sin x)/x)^2

Maclaurin series expansion of 1 - ((sin x)/x)^2

The Maclaurin series expansion of 1 - ((sin x)/x)^2 is

$1-\left({\frac {\sin x}{x}}\right)^{2}={\frac {x^{2}}{3}}-{\frac {2x^{4}}{45}}+{\frac {x^{6}}{315}}-{\frac {2x^{8}}{14175}}+{\frac {2x^{10}}{467775}}-{\frac {4x^{12}}{42567525}}+{\frac {x^{14}}{638512875}}-{\frac {2x^{16}}{97692469875}}+{\frac {2x^{18}}{9280784638125}}-{\frac {4x^{20}}{2143861251406875}}+\cdots \,$

Numerators of Maclaurin series for 1 - ((sin x)/x)^2

A048896 2 A000120() − 1, $\scriptstyle n\,\geq \,1\,$. Maximal power of 2 dividing $\scriptstyle n\,$ th Catalan number (A000108).

{1, 2, 1, 2, 2, 4, 1, 2, 2, 4, 2, 4, 4, 8, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 2, 4, 4, ...}

A000120 1's-counting sequence: number of 1's in binary expansion of $\scriptstyle n\,$ (or the binary weight of $\scriptstyle n\,\geq \,2\,$).

{1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, ...}

Denominators of Maclaurin series for 1 - ((sin x)/x)^2

A117972 Numerator of $\scriptstyle \zeta '(-2n),\,n\,\geq \,2\,$.

{3, -45, 315, -14175, 467775, -42567525, 638512875, -97692469875, 9280784638125, -2143861251406875, 147926426347074375, -48076088562799171875, 9086380738369043484375, -3952575621190533915703125, ...}

See also

• Montgomery's pair correlation conjecture