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inttrans

 fouriercos
 Fourier Cosine transform

 Calling Sequence fouriercos(expr, t, s)

Parameters

 expr - expression, equation, or set of expressions and/or equations to be transformed t - variable expr is transformed with respect to t s - parameter of transform opt - option to run this under (optional)

Description

 • The fouriercos function computes the Fourier Cosine transform (F(s)) of expr (f(t)), a linear transformation from $C\left[0,\infty \right)\to C\left[0,\infty \right)$ defined by:

$F\left(s\right)=\frac{\sqrt{2}\left({\int }_{0}^{\mathrm{\infty }}f\left(t\right)\mathrm{cos}\left(st\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt\right)}{\sqrt{\mathrm{\pi }}}$

 • The function $F\left(s\right)$ returned is defined on the positive real axis only.
 • Expressions involving exponentials, rational polynomials, trigonometrics (sin, cos) with linear arguments, and a variety of other functions can be transformed.
 • The Fourier Cosine transform is self-inverting.
 • The fouriercos function recognizes derivatives (diff or Diff) of functions of rapid descent and can be used to solve ODEs and PDEs.
 • The fouriercos function attempts to simplify an expression according to a set of heuristics and then match the result with a table of patterns. Entries can be added to this table by addtable(fouriercos, f(t), F(s), t, s), where F(s) is the transform of f(t), which may have an arbitrary number of parameters.
 • If the option opt is set to 'NO_INT', then the program will not resort to integration of the original problem if all other methods fail.  This will increase the speed at which the transform will run.
 • fouriercos  recognizes the Dirac-delta (or unit-impulse) function as Dirac(t) and Heaviside's unit step function as Heaviside(t).
 • The command with(inttrans,fouriercos) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{inttrans}\right)$
 $\left[{\mathrm{addtable}}{,}{\mathrm{fourier}}{,}{\mathrm{fouriercos}}{,}{\mathrm{fouriersin}}{,}{\mathrm{hankel}}{,}{\mathrm{hilbert}}{,}{\mathrm{invfourier}}{,}{\mathrm{invhilbert}}{,}{\mathrm{invlaplace}}{,}{\mathrm{invmellin}}{,}{\mathrm{laplace}}{,}{\mathrm{mellin}}{,}{\mathrm{savetable}}\right]$ (1)
 > $\mathrm{fouriercos}\left(\frac{3}{t+a},t,w\right)$
 $\frac{{3}{}\sqrt{{2}}{}\left({-}{\mathrm{sin}}{}\left({a}{}{w}\right){}{\mathrm{Ssi}}{}\left({a}{}{w}\right){-}{\mathrm{cos}}{}\left({a}{}{w}\right){}{\mathrm{Ci}}{}\left({a}{}{w}\right)\right)}{\sqrt{{\mathrm{π}}}}$ (2)
 > $\mathrm{fouriercos}\left(\frac{1}{{t}^{2}+1},t,s\right)$
 $\frac{{1}}{{2}}{}\sqrt{{2}}{}\sqrt{{\mathrm{π}}}{}{{ⅇ}}^{{-}{s}}$ (3)
 > $\mathrm{fouriercos}\left(,s,x\right)$
 $\frac{{1}}{{{x}}^{{2}}{+}{1}}$ (4)
 > $\mathrm{fouriercos}\left(\frac{ⅆ}{ⅆx}f\left(x\right),x,w\right)$
 $\frac{{w}{}{\mathrm{fouriersin}}{}\left({f}{}\left({x}\right){,}{x}{,}{w}\right){}\sqrt{{\mathrm{π}}}{-}\sqrt{{2}}{}{f}{}\left({0}\right)}{\sqrt{{\mathrm{π}}}}$ (5)
 > $\mathrm{fouriercos}\left(f\left(t\right),t,u\right)$
 ${\mathrm{fouriercos}}{}\left({f}{}\left({t}\right){,}{t}{,}{u}\right)$ (6)
 > $\mathrm{fouriercos}\left(,u,s\right)$
 ${f}{}\left({s}\right)$ (7)
 > $F≔{∫}_{0}^{\mathrm{∞}}g\left(x\right)\mathrm{cos}\left(xs\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx:$
 > $\mathrm{fouriercos}\left(F+\frac{1}{\sqrt{s}},s,t\right)$
 $\frac{{1}}{{2}}{}\sqrt{{2}}{}\sqrt{{\mathrm{π}}}{}{g}{}\left({t}\right){+}\frac{{1}}{\sqrt{{t}}}$ (8)
 > $\mathrm{fouriercos}\left(t{ⅇ}^{-3t}\mathrm{sin}\left(t\right),t,w\right)$
 ${-}\frac{{6}{}\sqrt{{2}}{}\left({3}{}{{w}}^{{4}}{+}{16}{}{{w}}^{{2}}{-}{100}\right)}{\sqrt{{\mathrm{π}}}{}{\left({{w}}^{{2}}{-}{2}{}{w}{+}{10}\right)}^{{2}}{}{\left({{w}}^{{2}}{+}{2}{}{w}{+}{10}\right)}^{{2}}}$ (9)
 > $\mathrm{de}≔\frac{{ⅆ}^{2}}{ⅆ{t}^{2}}y\left(t\right)-y\left(t\right)=\mathrm{cos}\left(2t\right):$
 > $\mathrm{fouriercos}\left(\mathrm{de},t,s\right)$
 ${-}\frac{{{s}}^{{2}}{}{\mathrm{fouriercos}}{}\left({y}{}\left({t}\right){,}{t}{,}{s}\right){}\sqrt{{\mathrm{π}}}{+}\sqrt{{2}}{}{\mathrm{D}}{}\left({y}\right){}\left({0}\right)}{\sqrt{{\mathrm{π}}}}{-}{\mathrm{fouriercos}}{}\left({y}{}\left({t}\right){,}{t}{,}{s}\right){=}\frac{{1}}{{2}}{}\sqrt{{2}}{}\sqrt{{\mathrm{π}}}{}{\mathrm{Dirac}}{}\left({s}{-}{2}\right)$ (10)
 > $\mathrm{solve}\left(,'\mathrm{fouriercos}'\left(y\left(t\right),t,s\right)\right)$
 ${-}\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}\left({\mathrm{π}}{}{\mathrm{Dirac}}{}\left({s}{-}{2}\right){+}{2}{}{\mathrm{D}}{}\left({y}\right){}\left({0}\right)\right)}{\sqrt{{\mathrm{π}}}{}\left({{s}}^{{2}}{+}{1}\right)}$ (11)
 > $\mathrm{fouriercos}\left(,s,t\right)$
 ${-}\frac{{1}}{{5}}{}{\mathrm{cos}}{}\left({2}{}{t}\right){-}{\mathrm{D}}{}\left({y}\right){}\left({0}\right){}{{ⅇ}}^{{-}{t}}$ (12)
 > $\mathrm{subs}\left(y\left(t\right)=,\mathrm{de}\right)$
 $\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}{}\left({-}\frac{{1}}{{5}}{}{\mathrm{cos}}{}\left({2}{}{t}\right){-}{\mathrm{D}}{}\left({y}\right){}\left({0}\right){}{{ⅇ}}^{{-}{t}}\right){+}\frac{{1}}{{5}}{}{\mathrm{cos}}{}\left({2}{}{t}\right){+}{\mathrm{D}}{}\left({y}\right){}\left({0}\right){}{{ⅇ}}^{{-}{t}}{=}{\mathrm{cos}}{}\left({2}{}{t}\right)$ (13)
 > $\mathrm{simplify}\left(\right)$
 ${\mathrm{cos}}{}\left({2}{}{t}\right){=}{\mathrm{cos}}{}\left({2}{}{t}\right)$ (14)
 > $\mathrm{fouriercos}\left(\mathrm{BesselJ}\left(0,4t\right),t,s-3\right)$
 $\frac{\sqrt{{2}}{}{\mathrm{Heaviside}}{}\left({7}{-}{s}\right)}{\sqrt{{\mathrm{π}}}{}\sqrt{{-}{\left({s}{-}{3}\right)}^{{2}}{+}{16}}}$ (15)

Add an entry to the table

 > $\mathrm{addtable}\left(\mathrm{fouriercos},h\left(t\right),H\left(s\right),t,s\right):$
 > $\mathrm{fouriercos}\left(h\left(t\right),t,s\right)$
 ${H}{}\left({s}\right)$ (16)