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I want to know how many roots does the following equation have and how to solve them all.

$f(z)=\frac{\text{sech}(35.0937 x)}{\left(e^{70.1873 x}-1.\right) x} \left((0.\, +0.0235822 i) x^2 \exp \left(35.0937 x+(0.\, +0.0000253744 i) \sqrt{-1.91278\times 10^{12} x^2+(0.\, +4.54224\times 10^{10} i)}\right)-(0.\, +0.0117911 i) x^2+\text{8.73597$\grave{ }$*${}^{\wedge}$-9} x \sqrt{-1.91278\times 10^{12} x^2+(0.\, +4.54224\times 10^{10} i)}+e^{70.1873 x} \left(-(0.\, +0.0117911 i) x^2-\text{8.73597$\grave{ }$*${}^{\wedge}$-9} x \sqrt{-1.91278\times 10^{12} x^2+(0.\, +4.54224\times 10^{10} i)}-0.00049\right)-0.00049\right)$

From above equation, I know there is a pole $(z=0)$. Plotting the quadrant in the region $\{0<x<0.5$, $0<y<0.5\}$, it seems that there are a root near point $(0.15+i0.08)$, and the other roots near by and along with y-axis.

I try Newton method given $z_0=0.1+i0.1$, but get an irrelative answer $z1=0.3625 - i0.0604$.

enter image description here

f[z_]:=1/((-1.+E^(70.1873 z)) z) (-0.00049-(0.+0.0117911 I) z^2+(0.+0.0235822 I) E^(35.0937 z+(0.+0.0000253744 I) Sqrt[(0.+4.54224*10^10 I)-1.91278*10^12 z^2]) z^2+8.73597*10^-9 z Sqrt[(0.+4.54224*10^10 I)-1.91278*10^12 z^2]+E^(70.1873 z) (-0.00049-(0.+0.0117911 I) z^2-8.73597*10^-9 z Sqrt[(0.+4.54224*10^10 I)-1.91278*10^12 z^2])) Sech[35.0937 z];

quad[z_] := Module[{q},u = N[ComplexExpand[Re[z]]];v = N[ComplexExpand[Im[z]]];   If[NumberQ[z], 
If[u == 0  || v == 0, q = 0, 
 If[u*v > 0, If[u > 0, q = 1, q = 3], If[u > 0, q = 4, q = 2]]], 
q = ComplexInfinity];q];

ContourPlot[quad[f[x + I  y]], {x, 0, 0.5}, {y, 0, 0.5},FrameLabel -> {"Re", "Im"}]

z1=NestWhile[(# - f[#]/f'[#]) &, 0.1 + I 0.1, Abs[f[#]] > 10^-7 &]
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Tried this with your function $f(z)$:

 funn0[z_] := f[z];
 gun0[x_, y_] := funn0[x + I*y];
 rgun0[x_, y_] := Re[gun0[x, y]];
 igun0[x_, y_] := Im[gun0[x, y]];
 p2 = ContourPlot[{rgun0[x, y] == 0, igun0[x, y] == 0}, {x, -0.1, .1}, {y, 0, .5},ContourStyle -> {Red, {Dashed, Blue}, Black, Black}, MaxRecursion -> 5]

enter image description here]

Next I find one of the roots as

 zr=z/.FindRoot[funn0[z], {z, 0.01 + .2*I}]

Which gives

{z -> 0.0149369 + 0.201915 I}

Don't forget to cross check the found root, as

 funn0[zr]

Which gives

 1.57061*10^-18 - 1.61687*10^-18 I.

Hence the root is CORRECT!

Similarly, other roots can also be found with a suitable guess value.

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  • $\begingroup$ Thank you. It is helpful. $\endgroup$ – Sandals Apr 6 at 4:04
  • $\begingroup$ @Sandals if it is helpful, please accept the answer. $\endgroup$ – math Apr 6 at 5:50
  • $\begingroup$ Do you have any idea how roots does this equation have? In figure, it seems like that it has infinite roots near y-axis, however, it is hard to image that in equation for me. And any method can solve the all roots at less in a given region? like region {0.5>x,y>0} $\endgroup$ – Sandals Apr 6 at 7:57

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