Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 205147 by Ghisom last updated on 10/Mar/24

$$\mathrm{solve}\mathrm{for}z\in \mathbb{C}$$$$z\ln z=z-2$$

Answered by pi314 last updated on 10/Mar/24

$$z=e^y$$$$\iff {ye}^y=e^y-2$$$$(y-1)e^{y-1}=-2e^{-1}$$$$y-1=W(-2e^{-1})$$$$y=1+W(-2e^{-1})$$$$z=e^{1+W(-2e^{-1})}$$$$$$

Commented by Frix last updated on 10/Mar/24

$${\mathrm{What}}^{\prime }\mathrm{s}\mathrm{the}\mathrm{approximate}\mathrm{value}\mathrm{of}z?$$

Answered by Frix last updated on 10/Mar/24

$$\mathrm{Different}\mathrm{method}:$$$$z\ln z-z+2=0$$$$z(\ln z-1)+2=0$$$$z=r{\mathrm{e}}^{\mathrm{i}\theta }$$$$r{\mathrm{e}}^{\mathrm{i}\theta }(\ln r-1+\mathrm{i}\theta )+2=0$$$$\{\begin{array}{l}r(\cos \theta \ln r-\cos \theta -\theta \sin \theta )+2=0\\ \mathrm{i}r(\sin \theta \ln r+\theta \cos \theta -\sin \theta )=0\end{array}$$$$\left(2\right)\Rightarrow r={\mathrm{e}}^{1-\frac{\theta }{\tan \theta }}$$$$\left(1\right)2-{\mathrm{e}}^{1-\frac{\theta }{\tan \theta }}\frac{\theta }{\sin \theta }=0$$$$\theta \approx \pm 1.13267249760$$$$r\approx 1.59896034818$$$$z\approx .678344933205\pm 1.44793727304\mathrm{i}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com