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Question Number 219360    Answers: 0   Comments: 0

prove Σ_(k=−∞) ^( ∞) J_k (z)=1

$$\mathrm{prove}\:\underset{{k}=−\infty} {\overset{\:\infty} {\sum}}\:{J}_{{k}} \left({z}\right)=\mathrm{1} \\ $$

Question Number 219359    Answers: 0   Comments: 0

is Σ_(k=−∞) ^∞ J_ν (k)= Σ_(k=−∞) ^∞ Σ_(l=0) ^∞ (((−1)^l )/(l!(l+ν)!))((k/2))^(2l+ν) =Σ_(l=0) ^∞ Σ_(k=−∞) ^∞ (((−1)^l )/(l!(l+ν)!))((k/2))^(2l+k) ??

$$\mathrm{is}\:\underset{{k}=−\infty} {\overset{\infty} {\sum}}\:{J}_{\nu} \left({k}\right)= \\ $$$$\underset{{k}=−\infty} {\overset{\infty} {\sum}}\:\underset{{l}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{l}} }{{l}!\left({l}+\nu\right)!}\left(\frac{{k}}{\mathrm{2}}\right)^{\mathrm{2}{l}+\nu} =\underset{{l}=\mathrm{0}} {\overset{\infty} {\sum}}\:\underset{{k}=−\infty} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{l}} }{{l}!\left({l}+\nu\right)!}\left(\frac{{k}}{\mathrm{2}}\right)^{\mathrm{2}{l}+{k}} ?? \\ $$

Question Number 219358    Answers: 0   Comments: 0

Σ_(h=−∞) ^∞ J_ν (h)=?? , ν∈Z\{2Z}

$$\underset{{h}=−\infty} {\overset{\infty} {\sum}}{J}_{\nu} \left({h}\right)=??\:,\:\nu\in\mathbb{Z}\backslash\left\{\mathrm{2}\mathbb{Z}\right\} \\ $$

Question Number 219357    Answers: 1   Comments: 0

∫_0 ^( ∞) ((ln(z))/(z^2 +1)) dz

$$\int_{\mathrm{0}} ^{\:\infty} \:\:\:\frac{\mathrm{ln}\left({z}\right)}{{z}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{d}{z} \\ $$

Question Number 219356    Answers: 1   Comments: 0

∫_0 ^( ∞) ((1−cos^2 (z))/z^2 )e^(−zt) dz

$$\int_{\mathrm{0}} ^{\:\infty} \:\:\frac{\mathrm{1}−\mathrm{cos}^{\mathrm{2}} \left({z}\right)}{{z}^{\mathrm{2}} }{e}^{−{zt}} \:\mathrm{d}{z} \\ $$

Question Number 219355    Answers: 1   Comments: 0

∫_0 ^( ∞) ((sin(z))/z)e^(−zt) dz=??

$$\int_{\mathrm{0}} ^{\:\infty} \:\:\frac{\mathrm{sin}\left({z}\right)}{{z}}{e}^{−{zt}} \mathrm{d}{z}=?? \\ $$

Question Number 219354    Answers: 1   Comments: 0

∫_(−∞) ^( +∞) ze^(−z^3 ) dz=??

$$\int_{−\infty} ^{\:+\infty} \:\:{ze}^{−{z}^{\mathrm{3}} } \:\mathrm{d}{z}=?? \\ $$

Question Number 219352    Answers: 1   Comments: 0

Question Number 219351    Answers: 1   Comments: 0

Question Number 219350    Answers: 1   Comments: 0

∫ (1/(cos(u)+sin(u)+1)) du=??

$$\int\:\:\frac{\mathrm{1}}{\mathrm{cos}\left({u}\right)+\mathrm{sin}\left({u}\right)+\mathrm{1}}\:\mathrm{d}{u}=?? \\ $$

Question Number 219341    Answers: 1   Comments: 1

∫(dx/(1 + sin^3 x + cos^3 x))

$$\int\frac{{dx}}{\mathrm{1}\:+\:{sin}^{\mathrm{3}} {x}\:+\:{cos}^{\mathrm{3}} {x}} \\ $$

Question Number 219340    Answers: 1   Comments: 0

F^→ (x,y,z)=−(x/( (√(x^2 +y^2 +z^2 ))))e_1 ^→ −(y/( (√(x^2 +y^2 +z^2 ))))e_2 ^→ −(z/( (√(x^2 +y^2 +z^2 ))))e_3 ^→ x^2 +y^2 +z^2 =R^2 ∫∫_( S) F^→ ∙dS^→ =??

$$\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\left({x},{y},{z}\right)=−\frac{{x}}{\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} }}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} −\frac{{y}}{\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} }}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} −\frac{{z}}{\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} }}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} ={R}^{\mathrm{2}} \\ $$$$\int\int_{\:\boldsymbol{\mathcal{S}}} \:\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\centerdot\mathrm{d}\overset{\rightarrow} {\boldsymbol{\mathcal{S}}}=?? \\ $$

Question Number 219339    Answers: 4   Comments: 0

Question Number 219338    Answers: 0   Comments: 0

F^→ =−xe_1 ^→ −ye_2 ^→ −ze_3 ^→ S^→ (u,v)= { (((2+3sin(u))cos(v))),(((2+3sin(v))sin(u))),((3cos(u))) :} u∈[0,2π] , v∈[0,2π] ∫∫_( S) F^→ ∙dS^→ =???

$$\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}=−{x}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} −{y}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} −{z}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$$\overset{\rightarrow} {\boldsymbol{\mathcal{S}}}\left({u},{v}\right)=\begin{cases}{\left(\mathrm{2}+\mathrm{3sin}\left({u}\right)\right)\mathrm{cos}\left({v}\right)}\\{\left(\mathrm{2}+\mathrm{3sin}\left({v}\right)\right)\mathrm{sin}\left({u}\right)}\\{\mathrm{3cos}\left({u}\right)}\end{cases} \\ $$$${u}\in\left[\mathrm{0},\mathrm{2}\pi\right]\:,\:{v}\in\left[\mathrm{0},\mathrm{2}\pi\right] \\ $$$$\int\int_{\:\boldsymbol{\mathcal{S}}} \:\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\centerdot\mathrm{d}\overset{\rightarrow} {\boldsymbol{\mathcal{S}}}=??? \\ $$

Question Number 219337    Answers: 5   Comments: 0

Question Number 219336    Answers: 0   Comments: 0

Question Number 219333    Answers: 0   Comments: 0

F^→ (x,y,z)=−xye_1 ^→ +yze_2 ^→ −xye_3 ^→ S^→ (u,v) { (((2+v∙cos(u))sin(2πv))),((v∙cos(u))),(((2+v∙cos(u))cos(2πv)+2v−2)) :} u∈[−π,π] , v∈[0,(π/2)] ∫∫_( S) F^→ ∙dS^→ =??

$$\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\left({x},{y},{z}\right)=−{xy}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} +{yz}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} −{xy}\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$$\overset{\rightarrow} {\boldsymbol{\mathcal{S}}}\left({u},{v}\right)\begin{cases}{\left(\mathrm{2}+{v}\centerdot\mathrm{cos}\left({u}\right)\right)\mathrm{sin}\left(\mathrm{2}\pi{v}\right)}\\{{v}\centerdot\mathrm{cos}\left({u}\right)}\\{\left(\mathrm{2}+{v}\centerdot\mathrm{cos}\left({u}\right)\right)\mathrm{cos}\left(\mathrm{2}\pi{v}\right)+\mathrm{2}{v}−\mathrm{2}}\end{cases} \\ $$$${u}\in\left[−\pi,\pi\right]\:,\:{v}\in\left[\mathrm{0},\frac{\pi}{\mathrm{2}}\right] \\ $$$$\int\int_{\:\boldsymbol{\mathcal{S}}} \:\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\centerdot\mathrm{d}\overset{\rightarrow} {\boldsymbol{\mathcal{S}}}=?? \\ $$

Question Number 219323    Answers: 0   Comments: 0

f(s)=(1/(2π)) ∫ e^(−it(s−α)) dt ∫_0 ^( ∞) ∫_(−∞) ^( +∞) e^(−it(s−α)) e^(−sp) dtds=?

$${f}\left({s}\right)=\frac{\mathrm{1}}{\mathrm{2}\pi}\:\int\:\:{e}^{−\boldsymbol{{i}}{t}\left({s}−\alpha\right)} \:\mathrm{d}{t}\: \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \int_{−\infty} ^{\:+\infty} \:\:{e}^{−\boldsymbol{{i}}{t}\left({s}−\alpha\right)} {e}^{−{sp}} \mathrm{d}{t}\mathrm{d}{s}=? \\ $$

Question Number 219318    Answers: 1   Comments: 0

Question Number 219315    Answers: 2   Comments: 0

Σ_(n=0) ^∞ (((−1)^n )/((2n+1)^4 ))=?

$$ \\ $$$$\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{4}} }=? \\ $$

Question Number 219316    Answers: 1   Comments: 0

Prove; ∫^( ∞) _( 0) (1/(2^2^(⌊x⌋) + {x})) dx = ln2

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{Prove}; \\ $$$$\:\:\:\:\:\:\:\underset{\:\mathrm{0}} {\int}^{\:\infty} \frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}^{\lfloor\boldsymbol{{x}}\rfloor} } \:+\:\left\{{x}\right\}}\:{dx}\:=\:{ln}\mathrm{2}\:\:\: \\ $$

Question Number 219305    Answers: 1   Comments: 0

Prove; I_0 (x) =(1/π)∫_0 ^( π) e^( x cox(θ)) dθ ; x^2 I_0 ^(′′) (x) + xI′_0 (x) − x^2 I_0 (x) = 0;

$$ \\ $$$$\:\:\:\:\:\:{Prove}; \\ $$$$\:\:\:{I}_{\mathrm{0}} \left({x}\right)\:=\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}} ^{\:\pi} \:{e}^{\:{x}\:{cox}\left(\theta\right)} \:{d}\theta\:; \\ $$$$\:\:\:{x}^{\mathrm{2}} {I}_{\mathrm{0}} ^{''} \left({x}\right)\:+\:{xI}'_{\mathrm{0}} \left({x}\right)\:−\:{x}^{\mathrm{2}} {I}_{\mathrm{0}} \left({x}\right)\:=\:\mathrm{0}; \\ $$$$\: \\ $$

Question Number 219304    Answers: 0   Comments: 0

Question Number 219301    Answers: 0   Comments: 0

f(s)=(1/(2π)) ∫_(−∞) ^(+∞) e^(−iω(s−α)) dω ∫_0 ^∞ (1/(2π))[e^(−st) ∫_( −∞) ^( +∞) e^(−iω(s−α)) dω]ds=....?

$${f}\left({s}\right)=\frac{\mathrm{1}}{\mathrm{2}\pi}\:\int_{−\infty} ^{+\infty} \:\:{e}^{−\boldsymbol{{i}}\omega\left({s}−\alpha\right)} \mathrm{d}\omega\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{\mathrm{2}\pi}\left[{e}^{−{st}} \:\int_{\:−\infty} ^{\:+\infty} \:\:{e}^{−\boldsymbol{{i}}\omega\left({s}−\alpha\right)} \mathrm{d}\omega\right]\mathrm{d}{s}=....? \\ $$

Question Number 219293    Answers: 1   Comments: 7

Question Number 219289    Answers: 2   Comments: 1

if a,b,c ∈ Z , and a^2 + b^2 = c^2 , then 3∣(ab) = ?

$$ \\ $$$$\:\:\:\:{if}\:\:\:\:\:\:\:\:\:\:\:{a},{b},{c}\:\in\:\mathbb{Z}\:\:\:, \\ $$$$\:\:\:\:\:{and}\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:=\:\:{c}^{\mathrm{2}} \:\:\:, \\ $$$$\:\:\:\:\:\:\:\:{then}\:\:\:\:\:\:\mathrm{3}\mid\left({ab}\right)\:=\:? \\ $$$$ \\ $$

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