What are fixed points of the Fourier Transform

$bf{1.}$ A more complete list of particular self-reciprocal Fourier functions of the first kind, i.e. eigenfunctions of the cosine Fourier transform $sqrt{frac{2}{pi}}int_0^infty f(x)cos ax dx=f(a)$:

$1.$ $displaystyle e^{-x^2/2}$ (more generally $e^{-x^2/2}H_{2n}(x)$, $H_n$ is Hermite polynomial)

$2.$ $displaystyle frac{1}{sqrt{x}}$ $qquad$ $3.$ $displaystylefrac{1}{coshsqrt{frac{pi}{2}}x}$ $qquad$ $4.$ $displaystyle frac{cosh frac{sqrt{pi}x}{2}}{cosh sqrt{pi}x}$ $qquad$$5.$ $displaystylefrac{1}{1+2cosh left(sqrt{frac{2pi}{3}}xright)}$

$6.$ $displaystyle frac{coshfrac{sqrt{3pi}x}{2}}{2cosh left( 2sqrt{frac{pi}{3}} xright)-1}$ $qquad$ $7.$ $displaystyle frac{coshleft(sqrt{frac{3pi}{2}}xright)}{cosh (sqrt{2pi}x)-cos(sqrt{3}pi)}$ $qquad$ $8.$ $displaystyle cosleft(frac{x^2}{2}-frac{pi}{8}right)$

$9.$ $displaystylefrac{cos frac{x^2}{2}+sin frac{x^2}{2}}{coshsqrt{frac{pi}{2}}x}$ $qquad$ $10.$ $displaystyle sqrt{x}J_{-frac{1}{4}}left(frac{x^2}{2}right)$ $qquad$ $11.$ $displaystyle frac{sqrt[4]{a} K_{frac{1}{4}}left(asqrt{x^2+a^2}right)}{(x^2+a^2)^{frac{1}{8}}}$

$12.$ $displaystyle frac{x e^{-betasqrt{x^2+beta^2}}}{sqrt{x^2+beta^2}sqrt{sqrt{x^2+beta^2}-beta}}$$qquad$ $13.$ $displaystyle psileft(1+frac{x}{sqrt{2pi}}right)-lnfrac{x}{sqrt{2pi}}$, $psi$ is digamma function.

Examples $1-5,8-10$ are from the chapter about self-reciprocal functions in Titschmarsh's book "Introduction to the theory of Fourier transform". Examples $11$ and $12$ can be found in Gradsteyn and Ryzhik. Examples $6$ and $7$ are from this question What are all functions of the form $frac{cosh(alpha x)}{cosh x+c}$ self-reciprocal under Fourier transform?. Some other self-reciprocal functions composed of hyperbolic functions are given in Bryden Cais's paper On the transformation of infinite series. Discussion of $13$ can be found in Berndt's article.

$bf{2.}$ Self-reciprocal Fourier functions of the second kind, i.e. eigenfunctions of the sine Fourier transform $sqrt{frac{2}{pi}}int_0^infty f(x)sin ax dx=f(a)$:

$1.$ $displaystyle frac{1}{sqrt{x}}$ $qquad$ $2.$ $displaystyle xe^{-x^2/2}$ (and more generally $e^{-x^2/2}H_{2n+1}(x)$)

$3.$ $displaystyle frac{1}{e^{sqrt{2pi}x}-1}-frac{1}{sqrt{2pi}x}$ $qquad$ $4.$ $displaystyle frac{sinh frac{sqrt{pi}x}{2}}{cosh sqrt{pi}x}$ $qquad$ $5.$ $displaystyle frac{sinhsqrt{frac{pi}{6}}x}{2cosh left(sqrt{frac{2pi}{3}}xright)-1}$

$6.$ $displaystyle frac{sinh(sqrt{pi}x)}{cosh sqrt{2pi} x-cos(sqrt{2}pi)}$ $qquad$ $7.$ $displaystyle frac{sin frac{x^2}{2}}{sinhsqrt{frac{pi}{2}}x}$ $qquad$ $8.$ $displaystyle frac{xK_{frac{3}{4}}left(asqrt{x^2+a^2}right)}{(x^2+a^2)^{frac{3}{8}}}$

$9.$ $displaystyle frac{x e^{-betasqrt{x^2+beta^2}}}{sqrt{x^2+beta^2}sqrt{sqrt{x^2+beta^2}+beta}}$$qquad$ $10.$ $displaystyle sqrt{x}J_{frac{1}{4}}left(frac{x^2}{2}right)$$qquad$ $11.$ $displaystyle e^{-frac{x^2}{4}}I_{0}left(frac{x^2}{4}right)$

$12.$ $displaystyle sinleft(frac{3pi}{8}+frac{x^2}{4}right)J_{0}left(frac{x^2}{4}right)$$qquad$ $13.$ $displaystyle frac{sinh sqrt{frac{2pi}{3}}x}{cosh sqrt{frac{3pi}{2}}x}$

Examples $1-5,7$ can be found in Titschmarsh's book cited above. $8-12$ can be found in Gradsteyn and Ryzhik. $13$ is from Bryden Cais, On the transformation of infinite series, where more functions of this kind are given.

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