How many solutions ?

Dec 2015
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How many solutions does \(\displaystyle 3e^z -z=0 \) has in the disk \(\displaystyle |z|\leq 1\) ? z-complex .
 

topsquark

Math Team
May 2013
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How many solutions does \(\displaystyle 3e^z -z=0 \) has in the disk \(\displaystyle |z|\leq 1\) ? z-complex .
Let z = x + iy.

Then
\(\displaystyle 3 e^{x + iy} - (x + iy) = 0\)

Use Euler's exponential equation to get rid of the exponential, then separate the equation into real and complex parts. Solve the equations. (The x equation is annoying but can be done numerically using W|A or something.

-Dan
 
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Dec 2015
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I took this from a pdf , now I lost the solution of the problem since I cannot find the pdf.
Applying W-Lambert function may lead to a fast answer .