Question Number 222117 by klipto last updated on 18/Jun/25 $$\boldsymbol{\mathrm{problem}}\:\mathrm{1}.\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{given}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{integer}} \\ $$$$\boldsymbol{\mathrm{m}},\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{all}}\:\boldsymbol{\mathrm{triples}}\left(\boldsymbol{\mathrm{n}},\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}}\right)\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{integers}},\boldsymbol{\mathrm{with}} \\ $$$$\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{relatively}}\:\boldsymbol{\mathrm{prime}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{m}},\boldsymbol{\mathrm{which}}\:\boldsymbol{\mathrm{satisfy}} \\ $$$$\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} \right)^{\boldsymbol{\mathrm{m}}} =\left(\boldsymbol{\mathrm{xy}}\right)^{\boldsymbol{\mathrm{n}}} \\ $$$$\boldsymbol{\mathrm{hint}}:\boldsymbol{\mathrm{utilize}}\:\boldsymbol{\mathrm{AM\&GM}},\boldsymbol{\mathrm{diophantine}}\:\boldsymbol{\mathrm{eqn}} \\ $$$$\boldsymbol{\mathrm{KLIPTO}}−\boldsymbol{\mathrm{QUANTA}}−\boldsymbol{\mathrm{OOZY}} \\ $$…
Question Number 222095 by wewji12 last updated on 17/Jun/25 $$\mathrm{could}\:\:\mathrm{I}\:\mathrm{consider}\:\:{Y}_{\nu} \left({z}\right)=\mathrm{cot}\left(\nu\pi\right){J}_{\nu} \left({z}\right)−\mathrm{csc}\left(\nu\pi\right){J}_{−\nu} \left({z}\right) \\ $$$$\mathrm{as}\:\infty−\infty\:\mathrm{form}\:\mathrm{limit}\:\mathrm{when}\:\nu\in\mathbb{Z} \\ $$$$\mathrm{and}\:\mathrm{How}\:\mathrm{can}\:\mathrm{i}\:\mathrm{calculate} \\ $$$${Y}_{\nu} \left({z}\right)=\mathrm{cot}\left(\nu\pi\right){J}_{\nu} \left({z}\right)−\mathrm{csc}\left(\nu\pi\right){J}_{−\nu} \left({z}\right)…?? \\ $$$$\underset{\alpha\rightarrow\nu} {\mathrm{lim}}\:\frac{\mathrm{cot}^{\mathrm{2}}…
Question Number 222057 by wewji12 last updated on 16/Jun/25 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}^{\mathrm{1}} ×\mathrm{2}^{\mathrm{2}} ×\mathrm{3}^{\mathrm{3}} ……×{n}^{{n}} }{{n}^{\frac{\mathrm{1}}{\mathrm{2}}{n}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}{n}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{12}}} ×{e}^{−\frac{\mathrm{1}}{\mathrm{4}}{n}^{\mathrm{2}} } }=??? \\ $$$$\mathrm{Help}…. \\ $$$$\mathrm{i}\:\mathrm{can}'\mathrm{t}\:\mathrm{Solve}\:\mathrm{that}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{a}_{{n}}…
Question Number 222044 by wewji12 last updated on 15/Jun/25 $$\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{evaluate} \\ $$$$\int_{−\infty} ^{\:\:\infty} \:\:\frac{\mathrm{sin}\left({z}+\mathrm{1}\right)}{\left({z}+\mathrm{1}\right)\left({z}^{\mathrm{2}} +\mathrm{1}\right)}\:\mathrm{d}{z} \\ $$ Answered by vnm last updated on 15/Jun/25 $$…
Question Number 221981 by MrGaster last updated on 14/Jun/25 $$\mathrm{Prove}:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}^{\mathrm{3}} }{{e}^{\mathrm{2}\pi{n}} −\mathrm{1}}=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{8}} }{\mathrm{5120}\pi^{\mathrm{6}} }−\frac{\mathrm{1}}{\mathrm{240}}=\frac{\mathrm{1}}{\mathrm{80}}\left(\frac{\varpi}{\pi}\right)^{\mathrm{4}} −\frac{\mathrm{1}}{\mathrm{240}} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 221939 by Ismoiljon_008 last updated on 13/Jun/25 $$ \\ $$$$\:\:\:{sin}^{\mathrm{2}} \mathrm{1}°\:+\:{sin}^{\mathrm{2}} \mathrm{3}°\:+\:{sin}^{\mathrm{2}} \mathrm{5}°\:+\:…\:+\:{sin}^{\mathrm{2}} \mathrm{269}°\:=\:? \\ $$$$\:\:\:\mathcal{H}{elp}\:{me},\:\:{please} \\ $$$$ \\ $$ Commented by Rasheed.Sindhi…
Question Number 221919 by fantastic last updated on 12/Jun/25 $${Acable}\:{can}\:{stand}\:{a}\:{maximum}\:{weight}\:{of}\:\mathrm{25kg}\: \\ $$$${If}\:{the}\:{length}\:{of}\:{wire}\:{is}\:{halved}\:{what}\:{is}\:{the} \\ $$$${maximum}\:{weight}\:{thsn}\:{can}\:{be}\:{hung}\:{from}\:{it}?? \\ $$ Commented by mr W last updated on 12/Jun/25 $$\sim\:\mathrm{50}\:{kg}…
Question Number 221843 by wewji12 last updated on 11/Jun/25 $$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\:{x}^{\mathrm{2}} \mathrm{csc}^{\mathrm{2}} \left({x}\right)\mathrm{d}{x} \\ $$$$\int\:\:−\frac{\mathrm{4}{z}^{\mathrm{2}} }{\left({e}^{\boldsymbol{{i}}{z}} −{e}^{−\boldsymbol{{i}}{z}} \right)^{\mathrm{2}} }\:\mathrm{d}{z}=\int\:\:\frac{{z}^{\mathrm{2}} {e}^{\mathrm{2}\boldsymbol{{i}}{z}} }{\left({e}^{\mathrm{2}\boldsymbol{{i}}{z}} −\mathrm{1}\right)^{\mathrm{2}} }\:\mathrm{d}{z} \\…
Question Number 221786 by wewji12 last updated on 10/Jun/25 $$\:\mathrm{Express}\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:{z}^{{z}^{{z}^{\iddots} } } \:\mathrm{d}{z}\:\:\mathrm{closed}\:\mathrm{form}… \\ $$$${z}\upuparrows^{\infty} ={z}^{{z}^{{z}^{\iddots} } } \\ $$ Terms of Service…
Question Number 221721 by ahmed2025 last updated on 09/Jun/25 Answered by wewji12 last updated on 09/Jun/25 $$\frac{\mathrm{d}{y}}{\mathrm{dln}\left({t}\right)}=\frac{\frac{\mathrm{d}{y}}{\mathrm{d}{t}}}{\frac{\mathrm{d}\left\{\mathrm{ln}\left({t}\right)\right\}}{\mathrm{d}{t}}}\:\left(\mathrm{Warning}\:\:\frac{\mathrm{d}{y}}{\mathrm{d}{x}}\:\mathrm{is}\:\mathrm{NOT}\:\mathrm{Fraction}!!!\right) \\ $$$$\frac{\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\left\{{t}^{\mathrm{2}} \mathrm{ln}\left({t}\right)\right\}}{\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\left\{\mathrm{ln}\left({t}\right)\right\}}=\frac{\mathrm{2}{t}\mathrm{ln}\left({t}\right)+{t}}{\frac{\mathrm{1}}{{t}}}={t}^{\mathrm{2}} +\mathrm{2}{t}^{\mathrm{2}} \mathrm{ln}\left({t}\right)={t}^{\mathrm{2}} \left(\mathrm{1}+\mathrm{2ln}\left({t}\right)\right) \\ $$$$\frac{\mathrm{d}{y}\left({t}\right)}{\mathrm{d}\left\{\mathrm{ln}\left({t}\right)\right\}}={t}^{\mathrm{2}}…