Solve-for-z-C-e-z-ln-z-

Question Number 204417 by Frix last updated on 16/Feb/24

Solve for z \in C

e^{z} = \ln z

Answered by mr W last updated on 17/Feb/24

s a y z = {r e}^{θ i} = r (\cos θ + i \sin θ)

w i t h r, θ \in R a n d r ⩾ 0 .

e^{r (\cos θ + i \sin θ)} = \ln r + θ i

e^{r \cos θ} e^{(r \sin θ) i} = \ln r + θ i

e^{r \cos θ} [\cos (r \sin θ) + i \sin (r \sin θ)] = \ln r + θ i

{\begin{cases} e^{r \cos θ} \cos (r \sin θ) = \ln r \\ e^{r \cos θ} \sin (r \sin θ) = θ \end{cases}

o n e o f i n f i n i t e s o l u t i o n s i s :

θ \approx \pm 1.33724, r \approx 1.37456

Commented by Frix last updated on 17/Feb/24

I think there are only 2 solutions .

The problems are very similar :

x^{n} = c versus \sqrt[n]{x} = c

e^{x} = a + b i versus \ln x = a + b i

Commented by mr W last updated on 17/Feb/24

Commented by Frix last updated on 18/Feb/24

Wolframalpha gives the following solutions :

z = \ln z \Rightarrow z = e^{- W (- 1)} \lor z = e^{- W_{- 1} (- 1)}

(z \approx . 318132 \pm 1 . 33724 i)

2 solutions

z = e^{z} \Rightarrow z = - W_{n} (- 1); n \in Z

infinitely many solutions

e^{z} = \ln z \Rightarrow z \approx . 318132 \pm 1 . 33724 i

2 solutions

Commented by mr W last updated on 17/Feb/24

e a c h o f t h e i n t e r s e c t i o n p o i n t s f r o m

t h e g r e e n c u r v e s a n d t h e r e d c u r v e s

r e p r e s e n t s a s o l u t i o n .

e . g . z = 2.4206 e^{\pm 6.9223 i}

Answered by Frix last updated on 17/Feb/24

e^{a} = b \Leftrightarrow a = \ln b

Let a = b = z

e^{z} = z \Leftrightarrow z = \ln z \Leftrightarrow e^{z} = \ln z

\Rightarrow We can solve

e^{z} = z and test our solutions

e^{z} = z

z = a + b i

e^{a} (\cos b + i \sin b) = a + b i

\sin b = b e^{- a}

\cos b = a e^{- a}

\Rightarrow \tan b = \frac{b}{a} \Leftrightarrow a = \frac{b}{\tan b}

\sin b = b e^{- \frac{b}{\tan b}} \Leftrightarrow b - e^{\frac{b}{\tan b}} \sin b = 0

b_{1} \approx \pm 1.33723570143 \Rightarrow a_{1} \approx . 318131505205

z_{1} \approx . 318131505205 \pm 1 . 33723570143 i

z_{1} \approx 1 . {37455701074 e}^{\pm 1 . 33723570143 i}

Test

e^{z_{1}} = z_{1} true

\ln z_{1} = z_{1} true

b_{2} \approx \pm 7.58863117847 \Rightarrow a_{2} \approx 2.06227772960

z_{2} \approx 2.06227772960 \pm 7 . 58863117847 i

z_{2} \approx 7 . {86386117609 e}^{\pm 1 . 30544587129 i}

Test

e^{z_{2}} = z_{2} true

\ln z_{2} \approx 2.06227772960 \pm 1 . 30544587129 i \neq z_{2} false

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