How-Can-derive-LambertW-z-in-the-Form-of-integral-W-z-1-pi-0-pi-ln-1-z-sin-t-t-e-t-cot-t-dt-z-1-e-Or-Similar-to-the-example-LambertW-z-How-other-Functions-can-be-Derived-

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Question Number 204573 by MathedUp last updated on 22/Feb/24

How Can derive LambertW(z) in the Form of integral??? W(z)=(1/π)∫_0 ^( π) ln(1+((z∙sin(t))/t)e^(t∙cot(t)) )dt , z∈[−(1/e),∞) Or Similar to the example.LambertW(z) How other Functions can be Derived in Integral Form

$$\mathrm{How}\:\mathrm{Can}\:\mathrm{derive}\:\mathrm{LambertW}\left({z}\right)\:\mathrm{in}\:\mathrm{the} \\ $$$$\:\mathrm{Form}\:\mathrm{of}\:\mathrm{integral}??? \\ $$$$\mathrm{W}\left({z}\right)=\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}} ^{\:\pi} \:\mathrm{ln}\left(\mathrm{1}+\frac{{z}\centerdot\mathrm{sin}\left({t}\right)}{{t}}{e}^{{t}\centerdot\mathrm{cot}\left({t}\right)} \right)\mathrm{d}{t}\:,\:{z}\in\left[−\frac{\mathrm{1}}{{e}},\infty\right) \\ $$$$\mathrm{Or}\:\mathrm{Similar}\:\mathrm{to}\:\mathrm{the}\:\mathrm{example}.\mathrm{LambertW}\left({z}\right) \\ $$$$\mathrm{How}\:\mathrm{other}\:\mathrm{Functions}\:\mathrm{can}\:\mathrm{be}\:\mathrm{Derived}\:\mathrm{in}\:\mathrm{Integral}\:\mathrm{Form} \\ $$

Commented by TonyCWX08 last updated on 22/Feb/24