$\mathrm{convert}\left(\mathrm{fourier}\left(f\left(t\right)\,t\,s\right)\,\mathrm{int}\right)$

${\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}f\left(t\right){\ⅇ}^{-\mathrm{I}ts}\ⅆt$

$\mathrm{convert}\left(\mathrm{invfourier}\left(f\left(t\right)\,t\,s\right)\,\mathrm{int}\right)$

$\frac{1}{2}\frac{{\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}f\left(t\right){\ⅇ}^{\mathrm{I}ts}\ⅆt}{\mathrm{\π}}$

$\mathrm{convert}\left(\mathrm{fouriercos}\left(f\left(t\right)\,t\,s\right)\,\mathrm{int}\right)$

$\frac{\sqrt{2}\left({\∫}_0^{\mathrm{\∞}}f\left(t\right)\cos \left(ts\right)\ⅆt\right)}{\sqrt{\mathrm{\π}}}$

$\mathrm{convert}\left(\mathrm{fouriersin}\left(f\left(t\right)\,t\,s\right)\,\mathrm{int}\right)$

$\frac{\sqrt{2}\left({\∫}_0^{\mathrm{\∞}}f\left(t\right)\sin \left(ts\right)\ⅆt\right)}{\sqrt{\mathrm{\π}}}$

$f:=\mathrm{randpoly}\left(\left[t\+a\,t\right]\,\mathrm{terms}\=3\right)$

$f:=87t^2-56{\left(t\+\mathrm{a~}\right)}^2{t}^2$

$\mathrm{fourier}\left(f\,t\,p\right)$

$2\mathrm{\π}\left(-56\mathrm{Dirac}\left(4\,p\right)\+112\mathrm{I}\mathrm{a~}\mathrm{Dirac}\left(3\,p\right)-87\mathrm{Dirac}\left(2\,p\right)\+56\mathrm{Dirac}\left(2\,p\right){\mathrm{a~}}^2\right)$

$\mathrm{invfourier}\left(\,p\,t\right)$

$-t^2\left(-87\+56t^2\+112t\mathrm{a~}\+56{\mathrm{a~}}^2\right)$

$\mathrm{fouriercos}\left(f\,t\,p\right)$

$\left(-\frac{87}{2}\sqrt{2}\sqrt{\mathrm{\π}}\+28\sqrt{2}{\mathrm{a~}}^2\sqrt{\mathrm{\π}}\right)\mathrm{Dirac}\left(2\,p\right)-28\sqrt{2}\sqrt{\mathrm{\π}}\mathrm{Dirac}\left(4\,p\right)-\frac{672\mathrm{a~}\sqrt{2}}{\sqrt{\mathrm{\π}}{p}^4}$

$\mathrm{fouriercos}\left(\,p\,t\right)$

$\frac{87}{2}{t}^2-28t^2{\mathrm{a~}}^2-28t^4-112t^3\mathrm{a~}$

$\mathrm{fouriersin}\left(f\,t\,p\right)$

$-56\sqrt{2}\sqrt{\mathrm{\π}}\mathrm{a~}\mathrm{Dirac}\left(3\,p\right)-\frac{174\sqrt{2}}{\sqrt{\mathrm{\π}}{p}^3}-\frac{56\sqrt{2}\left(-2{\mathrm{a~}}^2{p}^2\+24\right)}{\sqrt{\mathrm{\π}}{p}^5}$

$\mathrm{fouriersin}\left(\,p\,t\right)$

$-56t^3\mathrm{a~}\+87t^2-56t^2{\mathrm{a~}}^2-56t^4$

$\mathrm{fourier}\left({\ⅇ}^{-\mathrm{\λ}x}\mathrm{sech}\left(a{\ⅇ}^{-x}\right)\,x\,y\right)$

${\mathrm{a~}}^{-\mathrm{\λ~}-\mathrm{I}y}{2}^{1-2\mathrm{\λ~}-2\mathrm{I}y}\mathrm{\Γ}\left(\mathrm{\λ~}\+\mathrm{I}y\right)\left(\mathrm{\ζ}\left(0\,\mathrm{\λ~}\+\mathrm{I}y\,\frac{1}{4}\right)-\mathrm{\ζ}\left(0\,\mathrm{\λ~}\+\mathrm{I}y\,\frac{3}{4}\right)\right)$

$\mathrm{fouriercos}\left({\ⅇ}^{-ax}\,x\,y\right)$

$\frac{\sqrt{2}\mathrm{a~}}{\sqrt{\mathrm{\π}}\left({\mathrm{a~}}^2\+y^2\right)}$

$\mathrm{fouriersin}\left({\ⅇ}^{-ax}\,x\,y\right)$

$\frac{\sqrt{2}y}{\sqrt{\mathrm{\π}}\left({\mathrm{a~}}^2\+y^2\right)}$

$\mathrm{fourier}\left(\ln \left(\frac{1}{\sqrt{1\+x^2}}\right)\,x\,y\right)$

$\frac{\mathrm{\π}\left(-{\ⅇ}^y\mathrm{Heaviside}\left(-y\right)\+{\ⅇ}^{-y}\mathrm{Heaviside}\left(y\right)\right)}{y}$

$\mathrm{fouriercos}\left(\frac{1\ln \left(\left|\frac{a\+x}{a-x}\right|\right)}{x}\,x\,y\right)$

$-\sqrt{2}\sqrt{\mathrm{\π}}\mathrm{Ssi}\left(\mathrm{a~}y\right)$

$\mathrm{fouriersin}\left({\ⅇ}^{-x}\ln \left(x\right)\,x\,y\right)$

$\frac{\sqrt{2}\left(\arctan \left(y\right)-\mathrm{\γ}y-\frac{1}{2}y\ln \left(y^2\+1\right)\right)}{\sqrt{\mathrm{\π}}\left(y^2\+1\right)}$

$\mathrm{fourier}\left(\sin \left(ax\right)\,x\,y\right)$

$\mathrm{I}\mathrm{\π}\left(-\mathrm{Dirac}\left(y-\mathrm{a~}\right)\+\mathrm{Dirac}\left(y\+\mathrm{a~}\right)\right)$

$\mathrm{fouriercos}\left(\frac{\cos \left(x^2\right)}{x}\,x\,y\right)$

${\∫}_y^{\mathrm{\∞}}\frac{\sqrt{2}\left(\sin \left(\frac{1}{4}{\mathrm{\_U}}^2\right)\mathrm{FresnelC}\left(\frac{1}{8}\sqrt{2}\sqrt{\mathrm{\π}}{\mathrm{\_U}}^2\right)-\cos \left(\frac{1}{4}{\mathrm{\_U}}^2\right)\mathrm{FresnelS}\left(\frac{1}{8}\sqrt{2}\sqrt{\mathrm{\π}}{\mathrm{\_U}}^2\right)\right)}{\sqrt{\mathrm{\π}}}\ⅆ\mathrm{\_U}$

$\mathrm{fouriersin}\left(\frac{\cos \left(x^2\right)}{x}\,x\,y\right)$

$\frac{1}{2}\sqrt{2}\sqrt{\mathrm{\π}}\left(\frac{\sqrt{2}\mathrm{FresnelC}\left(\frac{1}{8}{y}^2\sqrt{2}\sqrt{\mathrm{\π}}\right)}{\sqrt{\mathrm{\π}}}\+\frac{\sqrt{2}\mathrm{FresnelS}\left(\frac{1}{8}{y}^2\sqrt{2}\sqrt{\mathrm{\π}}\right)}{\sqrt{\mathrm{\π}}}\right)$

$\mathrm{fouriercos}\left(\arccos \left(x\right)\mathrm{Heaviside}\left(1-x\right)\,x\,y\right)$

$\frac{1}{2}\frac{\sqrt{2}\sqrt{\mathrm{\π}}\mathrm{StruveH}\left(0\,y\right)}{y}$

$\mathrm{fouriersin}\left(\arcsin \left(x\right)\mathrm{Heaviside}\left(1-x\right)\,x\,y\right)$

$\frac{1}{2}\frac{\sqrt{2}\sqrt{\mathrm{\π}}\left(\mathrm{BesselJ}\left(0\,y\right)-\cos \left(y\right)\right)}{y}$

$\mathrm{fouriercos}\left(\frac{\tanh \left(ax\right)}{x}\,x\,y\right)$

$\frac{1}{2}\frac{\sqrt{2}\ln \left(\frac{\cosh \left(\frac{1}{2}\frac{y\mathrm{\π}}{\mathrm{a~}}\right)\+1}{\cosh \left(\frac{1}{2}\frac{y\mathrm{\π}}{\mathrm{a~}}\right)-1}\right)}{\sqrt{\mathrm{\π}}}$

$\mathrm{fouriersin}\left(\frac{\tanh \left(ax\right)}{x}\,x\,y\right)$

$\frac{\sqrt{2}\left(\frac{1}{2}\mathrm{\π}-\mathrm{I}\mathrm{lnGAMMA}\left(\frac{1}{2}\+\frac{\frac{1}{4}\mathrm{I}y}{\mathrm{a~}}\right)-\mathrm{I}\mathrm{lnGAMMA}\left(1-\frac{\frac{1}{4}\mathrm{I}y}{\mathrm{a~}}\right)\+\mathrm{I}\mathrm{lnGAMMA}\left(\frac{1}{2}-\frac{\frac{1}{4}\mathrm{I}y}{\mathrm{a~}}\right)\+\mathrm{I}\mathrm{lnGAMMA}\left(1\+\frac{\frac{1}{4}\mathrm{I}y}{\mathrm{a~}}\right)\right)}{\sqrt{\mathrm{\π}}}$

$\mathrm{fouriercos}\left(\mathrm{FresnelC}\left(\frac{1}{x}\right)\,x\,y\right)$

$\frac{1}{4}\frac{\sin \left(2\sqrt{\frac{\sqrt{2}y}{\sqrt{\mathrm{\π}}}}\right)\+\cos \left(2\sqrt{\frac{\sqrt{2}y}{\sqrt{\mathrm{\π}}}}\right)-{\ⅇ}^{-2\sqrt{\frac{\sqrt{2}y}{\sqrt{\mathrm{\π}}}}}}{y}$

$\mathrm{fouriersin}\left(\mathrm{FresnelS}\left(\frac{a}{x}\right)\,x\,y\right)$

$\frac{1}{4}\frac{2-{\ⅇ}^{-2\sqrt{\frac{\mathrm{a~}\sqrt{2}y}{\sqrt{\mathrm{\π}}}}}-\cos \left(2\sqrt{\frac{\mathrm{a~}\sqrt{2}y}{\sqrt{\mathrm{\π}}}}\right)-\sin \left(2\sqrt{\frac{\mathrm{a~}\sqrt{2}y}{\sqrt{\mathrm{\π}}}}\right)}{y}$

$\mathrm{fouriercos}\left(\mathrm{Ei}\left(ax\right)\,x\,y\right)$

$\frac{\sqrt{2}\arctan \left(\frac{y}{\mathrm{a~}}\right)}{\sqrt{\mathrm{\π}}y}$

$\mathrm{fouriersin}\left(\mathrm{Ssi}\left(-ax\right)\,x\,y\right)$

$\frac{1}{2}\frac{\sqrt{2}\sqrt{\mathrm{\π}}\mathrm{Heaviside}\left(y-\mathrm{a~}\right)}{y}-\frac{\sqrt{2}\sqrt{\mathrm{\π}}}{y}$

$\mathrm{fouriercos}\left(\frac{\mathrm{erf}\left(xa\right)}{x}\,x\,y\right)$

$-\frac{1}{2}\frac{\sqrt{2}\mathrm{Ei}\left(-\frac{1}{4}\frac{y^2}{{\mathrm{a~}}^2}\right)}{\sqrt{\mathrm{\π}}}$

$\mathrm{fouriersin}\left(\mathrm{erf}\left(x\right)\,x\,y\right)$

$\frac{\sqrt{2}\left(\frac{1}{y}-\frac{1-{\ⅇ}^{-\frac{1}{4}{y}^2}}{y}\right)}{\sqrt{\mathrm{\π}}}$

$\mathrm{fourier}\left(\mathrm{BesselJ}\left(0\,x\right)\,x\,y\right)$

$\frac{2\left(\mathrm{Heaviside}\left(1\+y\right)-\mathrm{Heaviside}\left(y-1\right)\right)}{\sqrt{1-y^2}}$

$\mathrm{fouriercos}\left(\mathrm{BesselY}\left(0\,a{x}^2\right)\,x\,y\right)$

$-\frac{1}{8}\frac{\sqrt{2}\sqrt{\mathrm{\π}}y\left({\mathrm{BesselJ}\left(\frac{1}{4}\,\frac{1}{8}\frac{y^2}{\mathrm{a~}}\right)}^2\+{\mathrm{BesselJ}\left(-\frac{1}{4}\,\frac{1}{8}\frac{y^2}{\mathrm{a~}}\right)}^2\right)}{\mathrm{a~}}$

$\mathrm{fouriersin}\left(\mathrm{BesselK}\left(0\,ax\right)\,x\,y\right)$

$\frac{\sqrt{2}\ln \left(\frac{y}{\mathrm{a~}}\+\sqrt{1\+\frac{y^2}{{\mathrm{a~}}^2}}\right)}{\sqrt{\mathrm{\π}}\sqrt{{\mathrm{a~}}^2-y^2}}$