I have shown that the following transcendental equation has only one real root. But I am looking for the argument to show that it has infinitely many numbers of roots in the complex plane. Moreover, these roots are countable. $$e^x+x=0$$
And real root dominate all other roots in the sense of absolute value. Or real part of all complex roots is less than the real root.
Would be grateful for the hint and remarks.
On
The solutions of $z+e^z=0$ come in complex conjugate pairs. Consider the roots with positive imaginary part. From $|z|=e^{Re(z)}$ one gets for $x=Re(z)\gg 1$ that $y=Im(z)\sim e^x\gg x$. Considering the dominance of the imaginary part, transform the equation to $$ -iz=ie^z=e^{z+i\pi/2}\implies z = Ln(-iz)+i2n\pi-i\frac\pi2 $$ for some $n\in\Bbb N$. Inserting this into itself gives in the next step $$ z= \ln((2n-\tfrac12)\pi)+i(2n-\tfrac12)\pi+w. $$ for some small $w$. Inserting back gives an equation for $w$, $$ w = Ln\left(1-i\frac{\ln((2n-\tfrac12)\pi)+w}{(2n-\tfrac12)\pi}\right) $$ For $n$ large enough this is a contraction on $\{w:|w|\le\frac12\}$, ensuring the existence of a solution. The same also holds for the unshifted iteration for $z$.
Different values of $n$ give different fixed-point iterations resulting in different solutions of the original equation, ensuring a countably infinite set of solutions.
Comparing the above first approximations with later iterates of the fixed-point iteration $$ z_{k+1} = Ln(-iz_k)+i(2n-\tfrac12)\pi $$ shows rapid (numerical) convergence and gives the table \begin{array}{l|lll} n& z_0 & z_{15} & z_{15}-z_0 \\ \hline 1 & (1.55019499396+4.71238898038j) & (1.53391331978+4.37518515309j) & (-0.0162816741736-0.337203827291j) \\ 2 & (2.39749285434+10.9955742876j) & (2.40158510487+10.7762995161j) & (0.00409225052324-0.219274771449j) \\ 3 & (2.84947797809+17.2787595947j) & (2.85358175541+17.1135355394j) & (0.00410377732121-0.165224055332j) \\ 4 & (3.15963290639+23.5619449019j) & (3.1629527388+23.4277475038j) & (0.00331983241242-0.134197398168j) \\ 5 & (3.39602168446+29.8451302091j) & (3.39869219676+29.7313107078j) & (0.00267051230882-0.113819501275j) \\ 6 & (3.58707692122+36.1283155163j) & (3.58926252453+36.0290217034j) & (0.00218560331097-0.0992938128549j) \\ 7 & (3.74741957129+42.4115008235j) & (3.74924254122+42.3231453612j) & (0.0018229699232-0.0883554622252j) \\ 8 & (3.88556990977+48.6946861306j) & (3.88711644955+48.6148985649j) & (0.00154653977456-0.0797875657055j) \\ 9 & (4.00693076678+54.9778714378j) & (4.00826205311+54.9049971233j) & (0.00133128633039-0.0728743144716j) \\ 10 & (4.11514435142+61.261056745j) & (4.116304664+61.193891332j) & (0.00116031258267-0.0671654130445j) \\ 11 & (4.21278282098+67.5442420522j) & (4.21380491472+67.48187952j) & (0.00102209373376-0.0623625321652j) \\ 12 & (4.301730307+73.8274273594j) & (4.3026389193+73.769167656j) & (0.000908612303842-0.0582597033192j) \\ \end{array} It also demonstrates a good fit of the initial approximation even for small $n$.
$f(z)=e^z+z$ is an entire function, hence it attains every complex value (with at most one exception) countable times, by Picard's theorem. Actually there are no exceptional values, since $$ f(z+2\pi i) = f(z)+2\pi i $$ (the presence of an exceptional value would imply the presence of infinite exceptional values) hence there are countable zeroes in the complex plane. The number of zeroes in the region $|z|\leq 2\pi M$ is given by $$ \frac{1}{2\pi i}\oint_{|z|=2\pi M}\frac{f'(z)}{f(z)}\,dz =\frac{1}{2\pi i}\oint_{|z|=2\pi M}\frac{1-z}{e^z+z}\,dz$$ but such roots are probably best accounted by noticing that $f(a+bi)=0$ is equivalent to
$$ \cos b = -a e^{-a}, \qquad \frac{\sin b}{b} = -e^{-a} $$ hence they are given by the intersections of the following blue/purple curves:
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The roots closest to the origin lie at $\approx 1.53391 \pm 4.37519i$.