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if-a-n-and-b-n-are-two-real-sequence-such-that-e-a-n-a-n-e-b-n-a-proof-that-a-n-gt-0-b-n-gt-0-b-if-a-n-gt-0-n-N-if-n-0-a-n-converge-then-n-0-b-n-a-n-converge-

Question Number 657 by 123456 last updated on 21/Feb/15 $${if}\:\left({a}_{{n}} \right)\:{and}\:\left({b}_{{n}} \right)\:{are}\:{two}\:{real}\:{sequence} \\ $$$${such}\:{that}\:{e}^{{a}_{{n}} } ={a}_{{n}} +{e}^{{b}_{{n}} } \\ $$$$\left.{a}\right)\:{proof}\:{that}\:{a}_{{n}} >\mathrm{0}\Rightarrow{b}_{{n}} >\mathrm{0} \\ $$$$\left.{b}\right)\:{if}\:{a}_{{n}} >\mathrm{0}\forall{n}\in\mathbb{N}\:{if}\:\underset{{n}=\mathrm{0}}…

find-all-root-by-demover-Z-4-2-2i-

Question Number 131713 by mohammad17 last updated on 07/Feb/21 $${find}\:{all}\:{root}\:{by}\:{demover}\:{Z}^{\mathrm{4}} =\mathrm{2}−\mathrm{2}{i} \\ $$ Answered by mathmax by abdo last updated on 07/Feb/21 $$\mathrm{2}−\mathrm{2i}\:=\mathrm{2}\left(\mathrm{1}−\mathrm{i}\right)\:=\mathrm{2}\sqrt{\mathrm{2}}\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}−\frac{\mathrm{i}}{\:\sqrt{\mathrm{2}}}\right)=\mathrm{2}\sqrt{\mathrm{2}}\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{4}}} \:=\mathrm{2}^{\frac{\mathrm{3}}{\mathrm{2}}} \:\mathrm{e}^{−\frac{\mathrm{i}\pi}{\mathrm{4}}}…

if-f-g-are-functions-of-R-R-not-constant-such-for-all-x-y-R-2-f-x-y-f-x-f-y-g-x-g-y-g-x-y-f-x-g-y-g-x-f-y-if-f-0-0-then-proof-os-disproof-that-x-R-f-x-2-g-x-2-1-

Question Number 636 by 123456 last updated on 17/Feb/15 $${if}\:{f},{g}\:{are}\:{functions}\:{of}\:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${not}\:{constant}\:{such}\:{for}\:{all}\:\left({x},{y}\right)\in\mathbb{R}^{\mathrm{2}} \\ $$$$\begin{cases}{{f}\left({x}+{y}\right)={f}\left({x}\right){f}\left({y}\right)−{g}\left({x}\right){g}\left({y}\right)}\\{{g}\left({x}+{y}\right)={f}\left({x}\right){g}\left({y}\right)+{g}\left({x}\right){f}\left({y}\right)}\end{cases} \\ $$$${if}\:{f}'\left(\mathrm{0}\right)=\mathrm{0}\:{then}\:{proof}\:{os}\:{disproof} \\ $$$${that}\:\forall{x}\in\mathbb{R},\left[{f}\left({x}\right)\right]^{\mathrm{2}} +\left[{g}\left({x}\right)\right]^{\mathrm{2}} =\mathrm{1} \\ $$ Commented by prakash…

e-3x-cosxdx-help-me-please-

Question Number 131706 by bounhome last updated on 07/Feb/21 $$\int{e}^{\mathrm{3}{x}} {cosxdx}=\:?\:{help}\:{me}\:{please}\: \\ $$ Answered by mathmax by abdo last updated on 07/Feb/21 $$\int\:\mathrm{e}^{\mathrm{3x}} \:\mathrm{cosx}\:\mathrm{dx}\:=\mathrm{Re}\left(\int\:\mathrm{e}^{\mathrm{3x}} \:\mathrm{e}^{\mathrm{ix}}…

Question-131684

Question Number 131684 by aurpeyz last updated on 07/Feb/21 Answered by ajfour last updated on 07/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\circleddash\left(−\mathrm{2}\mu{C}\:\right)\:\:\:\:\:\:\:\:\:\:\:\oplus\left(\mathrm{10}\mu{C}\right) \\ $$$$\frac{\mathrm{1}}{\mathrm{4}\pi\epsilon_{\mathrm{0}} }\frac{\mid{q}_{\mathrm{3}} {q}_{\mathrm{2}} \mid}{{r}^{\mathrm{2}} }=\frac{\mathrm{1}}{\mathrm{4}\pi\epsilon_{\mathrm{0}} }\frac{\mid{q}_{\mathrm{3}} {q}_{\mathrm{1}}…