Question Number 221416 by wewji12 last updated on 04/Jun/25 $$\mathrm{ex3}. \\ $$$$\mathrm{prove} \\ $$$${f}^{\left({n}\right)} \left(\alpha\right)=\frac{{n}!}{\mathrm{2}\pi\boldsymbol{{i}}}\:\oint_{\:\partial{S}} \:\frac{{f}\left({z}\right)}{\left({z}−\alpha\right)^{{n}+\mathrm{1}} }\:\mathrm{d}{z} \\ $$$$\mathrm{ex4}. \\ $$$$\mathrm{Let}\:{z}_{\mathrm{0}} \:\mathrm{be}\:\mathrm{any}\:\mathrm{point}\:\mathrm{interior}\:\mathrm{to}\:\mathrm{a}\:\mathrm{positively} \\ $$$$\mathrm{oriented}\:\mathrm{simple}\:\mathrm{closed}\:\mathrm{contour}\:\mathcal{C} \\…
Question Number 221415 by wewji12 last updated on 04/Jun/25 $$\int\:\:\mathrm{d}{z}\:\left[−\frac{\mathrm{1}}{\pi}\left(\frac{\mathrm{2}}{{z}}\right)^{\nu} \centerdot\underset{{k}=\mathrm{0}} {\overset{\nu−\mathrm{1}} {\sum}}\:\frac{\Gamma\left(\nu−{k}\right)}{{k}!}\left(\frac{{z}}{\mathrm{2}}\right)^{\mathrm{2}{k}} +\frac{\mathrm{2}}{\pi}\mathrm{ln}\left(\frac{\mathrm{1}}{\mathrm{2}}{z}\right){J}_{\nu} \left({z}\right)−\frac{\mathrm{1}}{\pi}\left(\frac{{z}}{\mathrm{2}}\right)^{\nu} \underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{k}} \left(\psi^{\left(\mathrm{0}\right)} \left({k}+\nu+\mathrm{1}\right)+\psi^{\left(\mathrm{0}\right)} \left({k}+\mathrm{1}\right)\right)}{{k}!\left({k}+\nu\right)!}\left(\frac{{z}}{\mathrm{2}}\right)^{\mathrm{2}{k}} \right] \\ $$ Answered…
Question Number 221391 by SdC355 last updated on 02/Jun/25 $$\int_{\mathrm{0}} ^{\:\infty} {J}_{\nu} ^{\left(\mathrm{1}\right)} \left({t}\right){Y}_{\nu} \left({t}\right)\mathrm{sin}\left({t}\right)\mathrm{d}{t}−\int_{\mathrm{0}} ^{\:\infty} {J}_{\nu} \left({t}\right){Y}_{\nu} ^{\left(\mathrm{1}\right)} \left({t}\right)\mathrm{sin}\left({t}\right)\mathrm{d}{t}=?? \\ $$$${J}_{\nu} \left({t}\right)\:\mathrm{is}\:\nu\:\mathrm{th}\:\mathrm{Bessel}\:\mathrm{function}\:\mathrm{first}\:\mathrm{Kind} \\ $$$${Y}_{\nu}…
Question Number 221387 by Nicholas666 last updated on 02/Jun/25 $$ \\ $$$$\:\:\:\:\:\:\:\underset{{k}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\left(\mathrm{2}\underset{{n}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} \:+\:{kn}}\right)^{\mathrm{2}} \:=\:? \\ $$$$ \\ $$ Answered by MrGaster…
Question Number 221380 by SdC355 last updated on 02/Jun/25 $$ \\ $$$$\mathrm{Problem}\:\mathrm{3}.\mathrm{11}\:\mathrm{Find}\:\mathrm{the}\:\mathrm{momentum}\:\mathrm{space}\:\mathrm{wave}\: \\ $$$$\mathrm{function}\:\boldsymbol{\Psi}\left({p},{t}\right)\:\mathrm{for}\:\mathrm{a}\:\mathrm{particle}\:\mathrm{in}\:\mathrm{the}\:\mathrm{ground}\:\mathrm{state}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{harmoic}\:\mathrm{oscillator}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{probability} \\ $$$$\left(\mathrm{to}\:\mathrm{two}\:\mathrm{signficant}\:\mathrm{digits}\right)\mathrm{that}\:\mathrm{a}\:\mathrm{measurement}\:\mathrm{of}\:\mathrm{on}\:\mathrm{a}\:\mathrm{particle}\: \\ $$$$\mathrm{in}\:\mathrm{this}\:\mathrm{state}\:\mathrm{would}\:\mathrm{yield}\:\mathrm{value}\:\mathrm{outside}\:\mathrm{the}\: \\ $$$$\mathrm{classical}\:\mathrm{range}\left(\mathrm{for}\:\mathrm{the}\:\mathrm{samenergy}\right) \\ $$$$\mathrm{Hint}\:\mathrm{Look}\:\mathrm{in}\:\mathrm{a}\:\mathrm{math}\:\mathrm{table}\:\mathrm{under}\:\mathrm{Normal}\:\mathrm{Distribution} \\…
Question Number 221382 by MrGaster last updated on 02/Jun/25 $$\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}^{−\mathrm{1}} \\ $$ Commented by mr W last updated on 05/Jun/25 $$=\frac{\mathrm{1}}{{n}!}\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{k}!\left({n}−{k}\right)!…
Question Number 221373 by Nicholas666 last updated on 01/Jun/25 Commented by MathematicalUser2357 last updated on 09/Jun/25 $$\frac{\frac{\int_{\mathrm{0}} ^{\infty} \mathrm{cos}\:\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{4}}{x}^{\mathrm{3}} {dx}}{\int_{\mathrm{0}} ^{\infty} \mathrm{sin}\:\mathrm{16}{x}^{\mathrm{3}} {dx}}+\left(\int_{\mathrm{0}} ^{\infty} \mathrm{ln}\left(\frac{\left(\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}\sqrt{\mathrm{2}}}…
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Question Number 221367 by SdC355 last updated on 01/Jun/25 $$\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \:\:\frac{\mathrm{1}}{\mathrm{5}−\mathrm{4sin}\left(\theta\right)}\:\mathrm{d}\theta=?? \\ $$$$\left(\mathrm{Complex}\:\mathrm{integral}\:\mathrm{method}\right) \\ $$ Answered by vnm last updated on 03/Jun/25 $$\int_{\mathrm{0}} ^{\mathrm{2}\pi}…