For an analytic function $f$ of z, Analytic attempts to find all complex zeros of $f\left(z\right)$ within the rectangular region $a<=\mathrm{Re}\left(z\right)<=b$, $c<=\mathrm{Im}\left(z\right)<=d$ in the complex plane. It may also find some zeros outside but close to the boundary of the region.

Setting of infolevel[RootFinding:-Analytic] to be between 1 and 7 will result in detailed information concerning the solving process being displayed.

The AnalyticZerosFound() calling sequence returns a sequence of the zeros which have been located. These may be accessed after Analytic returns, or if its computation is interrupted.

$\mathrm{with}\left(\mathrm{RootFinding}\right)$

$\left[\mathrm{Analytic}\&comma;\mathrm{AnalyticZerosFound}\&comma;\mathrm{BivariatePolynomial}\&comma;\mathrm{EnclosingBox}\&comma;\mathrm{EvaluateAtRoot}\&comma;\mathrm{HasRealRoots}\&comma;\mathrm{Homotopy}\&comma;\mathrm{Isolate}\&comma;\mathrm{NextZero}\&comma;\mathrm{Parametric}\&comma;\mathrm{RefineRoot}\&comma;\mathrm{WitnessPoints}\right]$

$f≔\tan \left(\sin \left(x\right)\right)-1$

$f≔\tan \left(\sin \left(x\right)\right)-1$

$\mathrm{Analytic}\left(f\&comma;x\&comma;-\frac{\mathrm{I}}{2}..1+\mathrm{I}\right)$

$0.903339110766515$

$g≔23x^5+105x^4-10x^2+17x$

$g≔23x^5+105x^4-10x^2+17x$

$\mathrm{Analytic}\left(g\&comma;x\&comma;\mathrm{re}=-5..1\&comma;\mathrm{im}=-1..1\right)$

$0.+0.\mathrm{I},\mathrm{-0.637181318531050},0.304066454284907-0.404061905751759\mathrm{I},0.304066454284907+0.404061905751759\mathrm{I},\mathrm{-4.53616898134312}$

$h≔10-\left(\ln \left(v+{\left(v^2-1\right)}^{\frac{1}{2}}\right)-\ln \left(3+{\left(3^2-1\right)}^{\frac{1}{2}}\right)\right)$

$h≔10-\ln \left(v+\sqrt{v^2-1}\right)+\ln \left(3+\sqrt{8}\right)$

$\mathrm{Analytic}\left(h\&comma;v\&comma;\mathrm{re}=2000..100000\&comma;\mathrm{im}=-100000..100000\&comma;\mathrm{digits}=32\right)$

$64189.825354267506015885528786917$

$\mathrm{Analytic}\left(\sin \left(x\right)\&comma;x\&comma;\mathrm{re}=-10..10\&comma;\mathrm{im}=-10..10\&comma;\mathrm{digits}=10\right)$

$0.+0.\mathrm{I},\mathrm{-6.283185305},\mathrm{-3.141592654},\mathrm{-9.424777960},6.283185305,3.141592654,9.424777960,0.+0.\mathrm{I},0.+0.\mathrm{I}$

$\mathrm{f0}≔\sin \left(x\right)-2^x$

$\mathrm{f0}≔\sin \left(x\right)-2^x$

$\mathrm{Analytic}\left(\mathrm{f0}\&comma;x\&comma;\mathrm{re}=-10..10\&comma;\mathrm{im}=-10..10\&comma;\mathrm{digits}=10\right)$

$\mathrm{-9.426231485},\mathrm{-3.247111097},\mathrm{-6.270228880},0.6634376855-1.171931328\mathrm{I},4.980776967-4.145343143\mathrm{I},9.224685705-7.087212710\mathrm{I},0.6634376855+1.171931328\mathrm{I},4.980776967+4.145343143\mathrm{I},9.224685705+7.087212710\mathrm{I}$

$\mathrm{numlist}≔\left[\frac{2}{3}\&comma;\frac{1+\mathrm{I}}{5}\&comma;-\frac{1}{2}-\frac{\mathrm{I}}{2}\&comma;7+\mathrm{I}\right]\&colon;$

$\mathrm{f1}≔\mathrm{mul}\left(z-\mathrm{z0}\&comma;\mathrm{z0}=\mathrm{numlist}\right)$

$\mathrm{f1}≔\left(z-\frac{2}{3}\right)\left(z-\frac{1}{5}-\frac{\mathrm{I}}{5}\right)\left(z+\frac{1}{2}+\frac{\mathrm{I}}{2}\right)\left(z-7-\mathrm{I}\right)$

$\mathrm{Analytic}\left(\mathrm{f1}\&comma;z\&comma;\mathrm{re}=-2..10\&comma;\mathrm{im}=-2..2\right)$

$0.666666666666665,0.200000000000000+0.200000000000000\mathrm{I},7.00000000000000+\mathrm{I},\mathrm{-0.50000000000000}-0.50000000000000\mathrm{I}$

$\mathrm{numlist}≔\left[\frac{2}{3}\&comma;\frac{1+\mathrm{I}}{5}\&comma;-\frac{1}{2}-\frac{\mathrm{I}}{2}\&comma;7+\mathrm{I}\right]\&colon;$

$\mathrm{f2}≔\mathrm{add}\left({\mathrm{z0}}^z\&comma;\mathrm{z0}=\mathrm{numlist}\right)$

$\mathrm{RootFinding}:-\mathrm{Analytic}\left(\mathrm{f2}\&comma;z\&comma;\mathrm{re}=-10..10\&comma;\mathrm{im}=-10..10\&comma;\mathrm{plot}\right)$

$\mathrm{RootFinding}:-\mathrm{AnalyticZerosFound}\left(\right)$

$\mathrm{f3}≔\mathrm{add}\left(\frac{1}{n^z}\&comma;n=1..10\right)$

$\mathrm{RootFinding}:-\mathrm{Analytic}\left(\mathrm{f3}\&comma;z\&comma;\mathrm{re}=-5..2\&comma;\mathrm{im}=-1..50\&comma;\mathrm{plot}\&comma;\mathrm{extra}\right)$

$\mathrm{f4}≔{\left(-100\mathrm{I}{x}^2+78x^4+39\mathrm{I}{x}^4-94x^3-17x\right)}^4+19x-73\mathrm{I}x-25-4\mathrm{I}-59x^5$

$\mathrm{zeros}≔\mathrm{RootFinding}:-\mathrm{Analytic}\left(\mathrm{f4}\&comma;x\&comma;\mathrm{re}=-10..10\&comma;\mathrm{im}=-10..10\right)$

$\mathrm{plots}\left[\mathrm{complexplot}\right]\left(\left\{\mathrm{zeros}\right\}\&comma;\mathrm{style}=\mathrm{point}\&comma;\mathrm{axes}=\mathrm{boxed}\right)$