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Question Number 218866    Answers: 0   Comments: 0

evaluate the following integral in closed form or express it in terms of known special functions; โˆซ_ ^โˆž K_(i๐›Œ) (at)J_๐›Ž (bt)^(๐›โˆ’1) dt where; โ€ขK_(i๐›Œ) (z) is the modified Bessel function of the second kind with imaginary order i๐›Œ, where ๐›ŒโˆˆR. โ€ข J_๐›Ž (z) is the Bessel function of the first kind with order ๐›Ž โˆˆC. โ€ข a,b are positif real constants. โ€ข ๐› is a complex parameter statisfying the conditions for the integral to converge.

$$ \\ $$$$\:\:\:{evaluate}\:{the}\:{following}\:{integral}\:{in}\:{closed}\:{form}\:{or}\:{express} \\ $$$$\:{it}\:{in}\:{terms}\:{of}\:{known}\:{special}\:{functions};\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\int_{ } ^{\infty} \boldsymbol{{K}}_{\boldsymbol{{i}\lambda}} \left(\boldsymbol{{at}}\right)\boldsymbol{{J}}_{\boldsymbol{\nu}} \left(\boldsymbol{{bt}}\right)^{\boldsymbol{\mu}โˆ’\mathrm{1}} \boldsymbol{{dt}} \\ $$$$\:\boldsymbol{{where}}; \\ $$$$\:\:\bullet\boldsymbol{{K}}_{\boldsymbol{{i}\lambda}} \left(\boldsymbol{{z}}\right)\:{is}\:{the}\:{modified}\:\boldsymbol{{B}}{essel}\:{function}\: \\ $$$$\:\:\:\:\:\:{of}\:{the}\:{second}\:{kind}\:{with}\:{imaginary}\:{order}\:\boldsymbol{{i}\lambda}, \\ $$$$\:\:\:\:\:\:\:{where}\:\boldsymbol{\lambda}\in\mathbb{R}. \\ $$$$\:\:\bullet\:\boldsymbol{{J}}_{\boldsymbol{\nu}} \left(\boldsymbol{{z}}\right)\:{is}\:{the}\:{Bessel}\:{function}\:{of}\:{the}\:{first}\:{kind}\:{with}\:{order}\:\boldsymbol{\nu}\:\in\mathbb{C}.\:\:\:\:\: \\ $$$$\:\:\bullet\:\boldsymbol{{a}},\boldsymbol{{b}}\:{are}\:{positif}\:{real}\:{constants}. \\ $$$$\:\:\bullet\:\boldsymbol{\mu}\:{is}\:{a}\:{complex}\:{parameter}\:{statisfying}\:{the}\:\:\: \\ $$$$\:\:\:\:\:\:\:{conditions}\:{for}\:{the}\:{integral}\:{to}\:{converge}.\:\:\:\: \\ $$$$ \\ $$$$\:\:\: \\ $$

Question Number 218857    Answers: 0   Comments: 0

((d )/dt) ((dx(t))/dt)โˆ’(x(t))^2 =k_0 ^2 ...?? how can i solve this Differantial Equation...???

$$\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\:\frac{\mathrm{d}{x}\left({t}\right)}{\mathrm{d}{t}}โˆ’\left({x}\left({t}\right)\right)^{\mathrm{2}} ={k}_{\mathrm{0}} ^{\mathrm{2}} ...?? \\ $$$$\mathrm{how}\:\mathrm{can}\:\mathrm{i}\:\mathrm{solve}\:\mathrm{this}\:\mathrm{Differantial}\:\mathrm{Equation}...??? \\ $$

Question Number 218855    Answers: 0   Comments: 0

(โˆš3)x^2 =(โˆš((64โˆ’x^2 )(x^2 โˆ’4)))+(โˆš((81โˆ’x^2 )(x^2 โˆ’1)))+(โˆš((49โˆ’x^2 )(x^2 โˆ’1)))solve

$$\sqrt{\mathrm{3}}{x}^{\mathrm{2}} =\sqrt{\left(\mathrm{64}โˆ’{x}^{\mathrm{2}} \right)\left({x}^{\mathrm{2}} โˆ’\mathrm{4}\right)}+\sqrt{\left(\mathrm{81}โˆ’{x}^{\mathrm{2}} \right)\left({x}^{\mathrm{2}} โˆ’\mathrm{1}\right)}+\sqrt{\left(\mathrm{49}โˆ’{x}^{\mathrm{2}} \right)\left({x}^{\mathrm{2}} โˆ’\mathrm{1}\right)}\boldsymbol{{solve}} \\ $$

Question Number 218853    Answers: 5   Comments: 0

Question Number 218854    Answers: 3   Comments: 0

Question Number 218850    Answers: 1   Comments: 0

Question Number 218849    Answers: 0   Comments: 0

Question Number 218848    Answers: 0   Comments: 0

Question Number 218846    Answers: 1   Comments: 0

Question Number 218844    Answers: 5   Comments: 0

Question Number 218845    Answers: 2   Comments: 0

Question Number 218836    Answers: 3   Comments: 0

Question Number 218835    Answers: 1   Comments: 0

Question Number 218834    Answers: 0   Comments: 0

Question Number 218833    Answers: 3   Comments: 0

Question Number 218832    Answers: 0   Comments: 0

Question Number 218975    Answers: 0   Comments: 0

In physics , Flux integral โˆฎ_( โˆ‚S) F^โ†’ โˆ™ dS^โ†’ is a concept that widely used in eletric equation or Heat Eqaution for example..... โˆฎ_( A) D^โ†’ โˆ™dA^โ†’ =Q_0 (Gauss law) D^โ†’ is displayment field โˆฎ_( S) B^โ†’ โˆ™dA^โ†’ =0 (Gauss law for magnetic) B^โ†’ is Magnetic field and in Heat Flux (โˆ‚E_(in) /โˆ‚t)โˆ’(โˆ‚E_(out) /โˆ‚t)=โˆฎ_( S) ๐›—_q ^โ†’ โˆ™dS^โ†’ But in mathematic it seems that Surface integral in the vector field is only extended version of the integral,why mathematic donโ€ฒt use surface integral like physics...??? i really curious

$$\mathrm{In}\:\mathrm{physics}\:,\:\mathrm{Flux}\:\mathrm{integral}\:\oint_{\:\partial\boldsymbol{\mathcal{S}}} \:\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}\centerdot\:\mathrm{d}\overset{\rightarrow} {\boldsymbol{\mathrm{S}}}\:\mathrm{is}\:\mathrm{a}\: \\ $$$$\mathrm{concept}\:\mathrm{that}\:\mathrm{widely}\:\mathrm{used}\:\mathrm{in}\:\mathrm{eletric}\:\mathrm{equation}\:\mathrm{or} \\ $$$$\mathrm{Heat}\:\mathrm{Eqaution} \\ $$$$\mathrm{for}\:\mathrm{example}.....\: \\ $$$$\oint_{\:{A}} \:\overset{\rightarrow} {\boldsymbol{\mathrm{D}}}\centerdot\mathrm{d}\overset{\rightarrow} {\boldsymbol{\mathrm{A}}}={Q}_{\mathrm{0}} \:\left(\mathrm{Gauss}\:\mathrm{law}\right)\:\overset{\rightarrow} {\boldsymbol{\mathrm{D}}}\:\mathrm{is}\:\mathrm{displayment}\:\mathrm{field} \\ $$$$\oint_{\:{S}} \:\overset{\rightarrow} {\boldsymbol{\mathrm{B}}}\centerdot\mathrm{d}\overset{\rightarrow} {\boldsymbol{\mathrm{A}}}=\mathrm{0}\:\left(\mathrm{Gauss}\:\mathrm{law}\:\mathrm{for}\:\mathrm{magnetic}\right)\:\overset{\rightarrow} {\boldsymbol{\mathrm{B}}}\:\mathrm{is}\:\mathrm{Magnetic}\:\mathrm{field} \\ $$$$\mathrm{and}\:\mathrm{in}\:\mathrm{Heat}\:\mathrm{Flux} \\ $$$$\frac{\partial\mathrm{E}_{\mathrm{in}} }{\partial{t}}โˆ’\frac{\partial\mathrm{E}_{\mathrm{out}} }{\partial{t}}=\oint_{\:{S}} \:\overset{\rightarrow} {\boldsymbol{\phi}}_{\mathrm{q}} \centerdot\mathrm{d}\overset{\rightarrow} {\boldsymbol{\mathrm{S}}}\:\: \\ $$$$\mathrm{But}\:\mathrm{in}\:\mathrm{mathematic}\:\mathrm{it}\:\mathrm{seems}\:\mathrm{that}\:\mathrm{Surface}\:\mathrm{integral} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{vector}\:\mathrm{field}\:\mathrm{is}\:\mathrm{only}\:\mathrm{extended}\:\mathrm{version}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{integral},\mathrm{why}\:\mathrm{mathematic}\:\mathrm{don}'\mathrm{t}\:\mathrm{use}\:\mathrm{surface}\:\mathrm{integral} \\ $$$$\mathrm{like}\:\mathrm{physics}...???\:\mathrm{i}\:\mathrm{really}\:\mathrm{curious}\: \\ $$

Question Number 218820    Answers: 1   Comments: 0

Question Number 218813    Answers: 3   Comments: 0

Question Number 218812    Answers: 5   Comments: 0

Question Number 218799    Answers: 0   Comments: 1

For those who are interested in cryptography. The below text has been encrypted using Vigenere cipher, such that numbers, punctuation marks and the letter E^(..) have remained the same. A keyword of length 9 has been used, which starts with the letter K. Decrypt the text.

$$\mathrm{For}\:\mathrm{those}\:\mathrm{who}\:\mathrm{are}\:\mathrm{interested}\:\mathrm{in}\:\mathrm{cryptography}. \\ $$$$\mathrm{The}\:\mathrm{below}\:\mathrm{text}\:\mathrm{has}\:\mathrm{been}\:\mathrm{encrypted}\:\mathrm{using} \\ $$$$\mathrm{Vigenere}\:\mathrm{cipher},\:\mathrm{such}\:\mathrm{that}\:\mathrm{numbers},\:\mathrm{punctuation} \\ $$$$\mathrm{marks}\:\mathrm{and}\:\mathrm{the}\:\mathrm{letter}\:\overset{..} {\mathrm{E}}\:\mathrm{have}\:\mathrm{remained}\:\mathrm{the}\:\mathrm{same}. \\ $$$$\mathrm{A}\:\mathrm{keyword}\:\mathrm{of}\:\mathrm{length}\:\mathrm{9}\:\mathrm{has}\:\mathrm{been}\:\mathrm{used},\:\mathrm{which} \\ $$$$\mathrm{starts}\:\mathrm{with}\:\mathrm{the}\:\mathrm{letter}\:\mathrm{K}.\:\mathrm{Decrypt}\:\mathrm{the}\:\mathrm{text}. \\ $$

Question Number 218997    Answers: 1   Comments: 0

Question Number 218792    Answers: 0   Comments: 0

Question Number 218785    Answers: 0   Comments: 0

Question Number 218781    Answers: 1   Comments: 0

prove: โˆซ_0 ^(ฯ€/4) arccos ((โˆš2)/( (โˆš(3โˆ’tan^2 x)))) dx=(ฯ€^2 /(24))

$$\mathrm{prove}: \\ $$$$\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\mathrm{arccos}\:\frac{\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{3}โˆ’\mathrm{tan}^{\mathrm{2}} \:{x}}}\:{dx}=\frac{\pi^{\mathrm{2}} }{\mathrm{24}} \\ $$

Question Number 218780    Answers: 2   Comments: 1

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