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Question Number 220502    Answers: 1   Comments: 0

each J_ν (z),Y_ν (z) are linear independent....?? W_(Ronskian) {J_ν ^ (z),Y_ν (z)}= determinant (((J_ν (z)),( Y_ν (z))),((J_ν ′(z)),(Y_ν ′(z)))) =J_ν ^((1)) (z)Y_ν (z)−J_ν (z)Y_ν ^((1)) (z) J_ν ^((1)) (z)Y_ν (z)=J_(ν−1) (z)Y_ν (z)−(ν/z)J_ν (z)Y_ν (z) J_ν (z)Y_ν ^((1)) (z)=Y_(ν−1) (z)J_ν (z)−(ν/z)J_ν (z)Y_ν (z) J_(ν−1) (z)Y_ν (z)−J_ν (z)Y_(ν−1) (z).... .....damn..... Result is (2/(πz)) ......

$$\mathrm{each}\:{J}_{\nu} \left({z}\right),{Y}_{\nu} \left({z}\right)\:\mathrm{are}\:\mathrm{linear}\:\mathrm{independent}....?? \\ $$$${W}_{\mathrm{Ronskian}} \left\{{J}_{\nu} ^{\:} \left({z}\right),{Y}_{\nu} \left({z}\right)\right\}=\begin{vmatrix}{{J}_{\nu} \left({z}\right)}&{\:{Y}_{\nu} \left({z}\right)}\\{{J}_{\nu} '\left({z}\right)}&{{Y}_{\nu} '\left({z}\right)}\end{vmatrix} \\ $$$$={J}_{\nu} ^{\left(\mathrm{1}\right)} \left({z}\right){Y}_{\nu} \left({z}\right)−{J}_{\nu} \left({z}\right){Y}_{\nu} ^{\left(\mathrm{1}\right)} \left({z}\right) \\ $$$${J}_{\nu} ^{\left(\mathrm{1}\right)} \left({z}\right){Y}_{\nu} \left({z}\right)={J}_{\nu−\mathrm{1}} \left({z}\right){Y}_{\nu} \left({z}\right)−\frac{\nu}{{z}}{J}_{\nu} \left({z}\right){Y}_{\nu} \left({z}\right) \\ $$$${J}_{\nu} \left({z}\right){Y}_{\nu} ^{\left(\mathrm{1}\right)} \left({z}\right)={Y}_{\nu−\mathrm{1}} \left({z}\right){J}_{\nu} \left({z}\right)−\frac{\nu}{{z}}{J}_{\nu} \left({z}\right){Y}_{\nu} \left({z}\right) \\ $$$${J}_{\nu−\mathrm{1}} \left({z}\right){Y}_{\nu} \left({z}\right)−{J}_{\nu} \left({z}\right){Y}_{\nu−\mathrm{1}} \left({z}\right).... \\ $$$$.....\mathrm{damn}..... \\ $$$$\mathrm{Result}\:\mathrm{is}\:\frac{\mathrm{2}}{\pi{z}}\:...... \\ $$

Question Number 220495    Answers: 0   Comments: 6

Question Number 220396    Answers: 0   Comments: 2

Question Number 220395    Answers: 2   Comments: 0

Find: Ω =Σ_(x=1) ^∞ Σ_(y=1) ^∞ (1/(x^2 y^3 (x^2 + 1)(y + 2))) = ?

$$\mathrm{Find}:\:\:\:\Omega\:=\underset{\boldsymbol{\mathrm{x}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\underset{\boldsymbol{\mathrm{y}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} \:\mathrm{y}^{\mathrm{3}} \:\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{1}\right)\left(\mathrm{y}\:+\:\mathrm{2}\right)}\:=\:? \\ $$

Question Number 220393    Answers: 1   Comments: 1

Question Number 220391    Answers: 3   Comments: 0

Question Number 220390    Answers: 1   Comments: 0

sinθ + sin(π + θ) + sin(2π + θ) + ... + sin(nπ + θ) = ? when n is an odd integer.

$$\mathrm{sin}\theta\:+\:\mathrm{sin}\left(\pi\:+\:\theta\right)\:+\:\mathrm{sin}\left(\mathrm{2}\pi\:+\:\theta\right)\:+\:...\: \\ $$$$+\:\mathrm{sin}\left({n}\pi\:+\:\theta\right)\:=\:?\:\mathrm{when}\:{n}\:\mathrm{is}\:\mathrm{an}\:\mathrm{odd} \\ $$$$\mathrm{integer}. \\ $$

Question Number 220388    Answers: 0   Comments: 1

Question Number 220380    Answers: 3   Comments: 0

lim_(n→∞) tan[(π/4)+(1/n)]^n =?

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}tan}\left[\frac{\pi}{\mathrm{4}}+\frac{\mathrm{1}}{{n}}\right]^{{n}} =? \\ $$

Question Number 220378    Answers: 0   Comments: 0

Question Number 220375    Answers: 1   Comments: 2

Question Number 220366    Answers: 2   Comments: 0

solve the system of equation using gaussian elimination method x+2y+3z=10 2x−3y+z=1 3x+y−2z=9

$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{system}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{equation}} \\ $$$$\boldsymbol{\mathrm{using}}\:\boldsymbol{\mathrm{gaussian}}\:\boldsymbol{\mathrm{elimination}}\:\boldsymbol{\mathrm{method}} \\ $$$$\boldsymbol{\mathrm{x}}+\mathrm{2}\boldsymbol{\mathrm{y}}+\mathrm{3}\boldsymbol{\mathrm{z}}=\mathrm{10} \\ $$$$\mathrm{2}\boldsymbol{\mathrm{x}}−\mathrm{3}\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}=\mathrm{1} \\ $$$$\mathrm{3}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}−\mathrm{2}\boldsymbol{\mathrm{z}}=\mathrm{9} \\ $$

Question Number 220365    Answers: 3   Comments: 0

Question Number 220362    Answers: 0   Comments: 0

Question Number 220377    Answers: 1   Comments: 0

Prove equation ∫_0 ^( ∞) f(u)g(u)e^(−uρ) du=(1/(2πi)) ∫_(−∞i+𝛄) ^( +∞i+𝛄) F(u)G(u−ρ)du F(u)=∫_0 ^( ∞) f(t)e^(−ut) dt G(u)=∫_0 ^( ∞) g(t)e^(−ut) dt

$$\mathrm{Prove}\:\mathrm{equation} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:{f}\left({u}\right)\mathrm{g}\left({u}\right){e}^{−{u}\rho} \mathrm{d}{u}=\frac{\mathrm{1}}{\mathrm{2}\pi\boldsymbol{{i}}}\:\int_{−\infty\boldsymbol{{i}}+\boldsymbol{\gamma}} ^{\:+\infty\boldsymbol{{i}}+\boldsymbol{\gamma}} \:\:{F}\left({u}\right){G}\left({u}−\rho\right)\mathrm{d}{u} \\ $$$${F}\left({u}\right)=\int_{\mathrm{0}} ^{\:\infty} \:{f}\left({t}\right){e}^{−{ut}} \mathrm{d}{t} \\ $$$${G}\left({u}\right)=\int_{\mathrm{0}} ^{\:\infty} \:\mathrm{g}\left({t}\right){e}^{−{ut}} \mathrm{d}{t} \\ $$

Question Number 220340    Answers: 2   Comments: 0

Question Number 220320    Answers: 1   Comments: 3

Question Number 220353    Answers: 1   Comments: 3

Question Number 220307    Answers: 1   Comments: 0

Question Number 220286    Answers: 3   Comments: 5

Question Number 220278    Answers: 0   Comments: 0

Question Number 220269    Answers: 1   Comments: 0

lim_(t→0) ((C_1 J_ν (t)+C_2 Y_ν (t)+H_ν (t))/(C_1 J_ν (t)+C_2 Y_ν (t)))=?? ν∈R J_ν (z) Bessel function First kind Y_ν (z) Bessel function Second Kind H_ν (z) Struve H function

$$\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{{C}_{\mathrm{1}} {J}_{\nu} \left({t}\right)+{C}_{\mathrm{2}} {Y}_{\nu} \left({t}\right)+\boldsymbol{\mathrm{H}}_{\nu} \left({t}\right)}{{C}_{\mathrm{1}} {J}_{\nu} \left({t}\right)+{C}_{\mathrm{2}} {Y}_{\nu} \left({t}\right)}=?? \\ $$$$\nu\in\mathbb{R} \\ $$$${J}_{\nu} \left({z}\right)\:\mathrm{Bessel}\:\mathrm{function}\:\mathrm{First}\:\mathrm{kind} \\ $$$${Y}_{\nu} \left({z}\right)\:\mathrm{Bessel}\:\mathrm{function}\:\mathrm{Second}\:\mathrm{Kind} \\ $$$$\boldsymbol{\mathrm{H}}_{\nu} \left({z}\right)\:\mathrm{Struve}\:\mathrm{H}\:\mathrm{function} \\ $$

Question Number 220266    Answers: 1   Comments: 0

2^a + 2^b + 2^c = 148

$$\mathrm{2}^{\mathrm{a}} \:\:+\:\:\mathrm{2}^{\mathrm{b}} \:\:+\:\:\mathrm{2}^{\mathrm{c}} \:\:=\:\:\mathrm{148} \\ $$

Question Number 220264    Answers: 1   Comments: 0

Question Number 220263    Answers: 3   Comments: 0

Question Number 220262    Answers: 7   Comments: 0

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