Question and Answers Forum

All Questions   Topic List

AllQuestion and Answers: Page 1383

Question Number 67647    Answers: 0   Comments: 0

Question Number 67631    Answers: 0   Comments: 3

Can you please tell me, where does this formula come from? And what means the factorial of a non- integer number? π = ((1/2)!)^2 × 4 I′ve verified the above equation with calculator. Thank you

$$ \\ $$$$ \\ $$$$\:\:\:\mathrm{Can}\:\mathrm{you}\:\mathrm{please}\:\mathrm{tell}\:\mathrm{me},\:\mathrm{where}\:\mathrm{does}\:\mathrm{this}\: \\ $$$$\:\:\:\mathrm{formula}\:\mathrm{come}\:\mathrm{from}? \\ $$$$\:\:\:\mathrm{And}\:\mathrm{what}\:\mathrm{means}\:\mathrm{the}\:\mathrm{factorial}\:\mathrm{of}\:\mathrm{a}\:\mathrm{non}- \\ $$$$\:\:\:\mathrm{integer}\:\mathrm{number}? \\ $$$$ \\ $$$$\:\:\:\:\:\:\pi\:=\:\left(\frac{\mathrm{1}}{\mathrm{2}}!\right)^{\mathrm{2}} ×\:\mathrm{4} \\ $$$$ \\ $$$$\:\:\:{I}'{ve}\:{verified}\:{the}\:{above}\:{equation}\:{with}\:{calculator}. \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Thank}\:\mathrm{you} \\ $$$$ \\ $$$$ \\ $$

Question Number 67628    Answers: 0   Comments: 0

∫x^n lnx/n^(x ) dx

$$\int{x}^{{n}} {lnx}/{n}^{{x}\:} \:{dx} \\ $$

Question Number 67623    Answers: 1   Comments: 5

Question Number 67618    Answers: 1   Comments: 0

find the area abovnded r=cos2θ

$${find}\:{the}\:{area}\:{abovnded}\:{r}={cos}\mathrm{2}\theta \\ $$

Question Number 67617    Answers: 0   Comments: 2

Question Number 67615    Answers: 2   Comments: 1

Question Number 67574    Answers: 0   Comments: 1

Solve x^2 +1<−5

$$\mathrm{Solve}\:\mathrm{x}^{\mathrm{2}} +\mathrm{1}<−\mathrm{5} \\ $$

Question Number 67572    Answers: 0   Comments: 1

∫_(−(π/2)) ^(π/2) {sin∣x∣+cos∣x∣} dx

$$\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \left\{{sin}\mid{x}\mid+{cos}\mid{x}\mid\right\}\:{dx} \\ $$

Question Number 67564    Answers: 0   Comments: 8

Question Number 67561    Answers: 1   Comments: 0

Question Number 67559    Answers: 0   Comments: 3

calculate Σ_(n=0) ^∞ (1/(n^(2 ) +1)) and Σ_(n=0) ^∞ (((−1)^n )/(n^2 +1))

$${calculate}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}\:} +\mathrm{1}}\:\:{and}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} \:+\mathrm{1}} \\ $$

Question Number 67542    Answers: 0   Comments: 4

let f(a) =∫_(−∞) ^(+∞) (dx/((x^2 +1)(a +e^(ix) ))) with a>0 1)find a explicit form of f(a) 2) determine also g(a)=∫_(−∞) ^(+∞) (dx/((x^2 +1)(a+e^(ix) )^2 )) 3)let I =Re(∫_(−∞) ^(+∞) (dx/((x^2 +1)(2+e^(ix) )))) and J=Im(∫_(−∞) ^(+∞) (dx/((x^2 +1)(2+e^x )))) determine I and J and its values.

$${let}\:{f}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({a}\:+{e}^{{ix}} \right)}\:\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{also}\:{g}\left({a}\right)=\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({a}+{e}^{{ix}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){let}\:{I}\:={Re}\left(\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{2}+{e}^{{ix}} \right)}\right)\:{and}\:{J}={Im}\left(\int_{−\infty} ^{+\infty} \:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{2}+{e}^{{x}} \right)}\right) \\ $$$$\:{determine}\:{I}\:{and}\:{J}\:\:{and}\:\:{its}\:{values}. \\ $$

Question Number 67540    Answers: 0   Comments: 2

prove that ∣Γ((1/2)+it)∣ =(√((2π)/(e^(πt) +e^(−πt) ))) and ∣Γ(1+it)∣ =(√((2πt)/(e^(πt) −e^(−πt) )))

$${prove}\:{that}\:\:\:\mid\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}+{it}\right)\mid\:=\sqrt{\frac{\mathrm{2}\pi}{{e}^{\pi{t}} \:+{e}^{−\pi{t}} }} \\ $$$${and}\:\mid\Gamma\left(\mathrm{1}+{it}\right)\mid\:=\sqrt{\frac{\mathrm{2}\pi{t}}{{e}^{\pi{t}} −{e}^{−\pi{t}} }} \\ $$

Question Number 67539    Answers: 0   Comments: 3

calculate ∫_0 ^∞ (du/(∣u+z∣^2 )) if z =r e^(iθ) and −π<θ<π

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{du}}{\mid{u}+{z}\mid^{\mathrm{2}} }\:\:{if}\:{z}\:={r}\:{e}^{{i}\theta} \:\:\:{and}\:−\pi<\theta<\pi \\ $$

Question Number 67538    Answers: 0   Comments: 2

prove that ((Γ^′ (z))/(Γ(z))) =−γ−(1/z) −Σ_(n=1) ^∞ ((1/(z+n))−(1/n))

$${prove}\:{that}\:\frac{\Gamma^{'} \left({z}\right)}{\Gamma\left({z}\right)}\:=−\gamma−\frac{\mathrm{1}}{{z}}\:−\sum_{{n}=\mathrm{1}} ^{\infty} \left(\frac{\mathrm{1}}{{z}+{n}}−\frac{\mathrm{1}}{{n}}\right) \\ $$

Question Number 67537    Answers: 0   Comments: 1

prove that (1/(Γ(z))) =z e^(γz) Π_(n=1) ^∞ (1+(z/n))e^(−(z/n))

$${prove}\:{that}\:\frac{\mathrm{1}}{\Gamma\left({z}\right)}\:={z}\:{e}^{\gamma{z}} \:\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}+\frac{{z}}{{n}}\right){e}^{−\frac{{z}}{{n}}} \\ $$

Question Number 67535    Answers: 1   Comments: 5

Question Number 67534    Answers: 0   Comments: 3

find the value of Π_(n=2) ^∞ ((n^3 −1)/(n^3 +1)) and Π_(n=1) ^∞ (1+(1/n^2 ))

$${find}\:{the}\:{value}\:{of}\:\:\prod_{{n}=\mathrm{2}} ^{\infty} \:\frac{{n}^{\mathrm{3}} −\mathrm{1}}{{n}^{\mathrm{3}} \:+\mathrm{1}} \\ $$$${and}\:\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right) \\ $$

Question Number 67533    Answers: 0   Comments: 0

find the value of Π_(n=1) ^∞ (1+(1/(n(n+2)))) Π_(n=1) ^∞ (1−(2/(n(n+1))))

$${find}\:{the}\:{value}\:{of}\:\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}+\frac{\mathrm{1}}{{n}\left({n}+\mathrm{2}\right)}\right) \\ $$$$\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}−\frac{\mathrm{2}}{{n}\left({n}+\mathrm{1}\right)}\right) \\ $$

Question Number 67532    Answers: 0   Comments: 1

prove that π cotan(πα) =lim_(n→+∞) Σ_(k=−n) ^n (1/(α−k))

$${prove}\:{that}\:\:\pi\:{cotan}\left(\pi\alpha\right)\:={lim}_{{n}\rightarrow+\infty} \:\:\:\sum_{{k}=−{n}} ^{{n}} \:\:\frac{\mathrm{1}}{\alpha−{k}} \\ $$

Question Number 67531    Answers: 0   Comments: 1

prove that cos(πz) =Π_(n=1) ^∞ (1−(z^2 /(((1/2)+n)^2 )))

$${prove}\:{that}\:{cos}\left(\pi{z}\right)\:=\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}−\frac{{z}^{\mathrm{2}} }{\left(\frac{\mathrm{1}}{\mathrm{2}}+{n}\right)^{\mathrm{2}} }\right) \\ $$

Question Number 67530    Answers: 0   Comments: 2

calculate ∫_0 ^∞ (x^(n−3) /(1+x^(2n) ))dx with n≥3

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{{n}−\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{2}{n}} }{dx}\:\:{with}\:{n}\geqslant\mathrm{3} \\ $$

Question Number 67528    Answers: 0   Comments: 3

calculate ∫_(−∞) ^(+∞) ((1+x^3 )/(1+x^6 ))dx

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{\mathrm{1}+{x}^{\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{6}} }{dx} \\ $$

Question Number 67527    Answers: 0   Comments: 1

calculate ∫_(−∞) ^(+∞) ((1+x^2 )/(1+x^4 ))dx

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\frac{\mathrm{1}+{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{4}} }{dx} \\ $$$$ \\ $$

Question Number 67526    Answers: 0   Comments: 1

find the value of ∫_0 ^(2π) (dx/(3+2sinx +cosx))

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dx}}{\mathrm{3}+\mathrm{2}{sinx}\:+{cosx}} \\ $$

  Pg 1378      Pg 1379      Pg 1380      Pg 1381      Pg 1382      Pg 1383      Pg 1384      Pg 1385      Pg 1386      Pg 1387   

Terms of Service

Privacy Policy

Contact: info@tinkutara.com