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Question Number 120554 Answers: 0 Comments: 0

Prove that for all a>0 ∫_([−a;a]) arg(Γ((1/2) −ix))dx =0 Deduce that f: x→arg(Γ((1/2) −ix)) is an old function on R

$${Prove}\:{that}\:{for}\:{all}\:\:{a}>\mathrm{0} \\ $$$$\int_{\left[−{a};{a}\right]} {arg}\left(\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\:−{ix}\right)\right){dx}\:=\mathrm{0} \\ $$$${Deduce}\:{that}\: \\ $$$${f}:\:{x}\rightarrow{arg}\left(\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\:−{ix}\right)\right)\:\:{is}\:{an}\:{old}\:{function}\:{on}\:\mathbb{R} \\ $$

Question Number 120553 Answers: 1 Comments: 0

∫_0 ^∞ ∣Γ((1/2) −ix)∣^2 dx = (π/2)

$$\:\: \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\infty} \mid\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\:−{ix}\right)\mid^{\mathrm{2}} {dx}\:=\:\frac{\pi}{\mathrm{2}}\: \\ $$

Question Number 120552 Answers: 0 Comments: 0

evaluate: ∫_0 ^( ∞) ((x/(x+1)))^(x!) dx

$$\: \\ $$$$\: \\ $$$$\:\:\:\:\:\:\:\mathrm{evaluate}:\:\int_{\mathrm{0}} ^{\:\infty} \left(\frac{{x}}{{x}+\mathrm{1}}\right)^{{x}!} {dx} \\ $$$$ \\ $$$$ \\ $$

Question Number 120551 Answers: 1 Comments: 0

(1/( (√3)−tan x)) − (1/( (√3)+tan x)) = sin 2x

$$\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}−\mathrm{tan}\:\mathrm{x}}\:−\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}+\mathrm{tan}\:\mathrm{x}}\:=\:\mathrm{sin}\:\mathrm{2x} \\ $$

Question Number 120549 Answers: 3 Comments: 0

Question Number 120545 Answers: 1 Comments: 2

((√x))^(x/( (√x))) = (1/( (√2))) x=?

$$\left(\sqrt{\boldsymbol{{x}}}\right)^{\frac{\boldsymbol{{x}}}{\:\sqrt{\boldsymbol{{x}}}}} \:=\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\:\:\:\:\boldsymbol{{x}}=? \\ $$

Question Number 120544 Answers: 1 Comments: 0

∫(1/((cos x)^6 ))=?

$$\int\frac{\mathrm{1}}{\left(\mathrm{cos}\:{x}\right)^{\mathrm{6}} }=? \\ $$

Question Number 120542 Answers: 0 Comments: 0

Question Number 120541 Answers: 1 Comments: 0

Question Number 120534 Answers: 2 Comments: 1

Question Number 120531 Answers: 1 Comments: 1

Question Number 120529 Answers: 1 Comments: 1

Question Number 120528 Answers: 0 Comments: 0

Let a>0 , g(x)=Σ_(n∈Z) exp(−(((x−n)^2 )/a)) Find the Fourier coefficients of g and deduce that for x∈R^∗ Σ_(n∈Z) exp(−((πn^2 )/x^2 ))= x .Σ_(n∈Z) exp(−πn^2 x^2 )

$${Let}\:\:{a}>\mathrm{0}\:,\:{g}\left({x}\right)=\underset{{n}\in\mathbb{Z}} {\sum}{exp}\left(−\frac{\left({x}−{n}\right)^{\mathrm{2}} }{{a}}\right) \\ $$$${Find}\:{the}\:{Fourier}\:{coefficients}\:{of}\:\:\:{g} \\ $$$${and}\:{deduce}\:{that}\:{for}\:{x}\in\mathbb{R}^{\ast} \\ $$$$\:\underset{{n}\in\mathbb{Z}} {\sum}{exp}\left(−\frac{\pi{n}^{\mathrm{2}} }{{x}^{\mathrm{2}} }\right)=\:{x}\:.\underset{{n}\in\mathbb{Z}} {\sum}{exp}\left(−\pi{n}^{\mathrm{2}} {x}^{\mathrm{2}} \right) \\ $$$$ \\ $$

Question Number 120524 Answers: 0 Comments: 0

1/.Given E=C([0,1],R) are mapping set [0,1]→R. (a). Show that E is vector space. (b).For f∈E ,let ∣∣f∣∣_1 =∫_0 ^1 ∣f(x)∣dx . Show that (E,∣∣.∣∣_1 ) are normed space. (Helpe me please)

$$\mathrm{1}/.\mathrm{Given}\:\mathrm{E}=\mathrm{C}\left(\left[\mathrm{0},\mathrm{1}\right],\mathbb{R}\right)\:\mathrm{are}\:\mathrm{mapping}\:\mathrm{set}\:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R}. \\ $$$$\left(\mathrm{a}\right).\:\mathrm{Show}\:\mathrm{that}\:\mathrm{E}\:\mathrm{is}\:\mathrm{vector}\:\mathrm{space}. \\ $$$$\left(\mathrm{b}\right).\mathrm{For}\:\:\mathrm{f}\in\mathrm{E}\:,\mathrm{let}\:\mid\mid\mathrm{f}\mid\mid_{\mathrm{1}} =\int_{\mathrm{0}} ^{\mathrm{1}} \mid\mathrm{f}\left(\mathrm{x}\right)\mid\mathrm{dx}\:. \\ $$$$\:\mathrm{Show}\:\mathrm{that}\:\left(\mathrm{E},\mid\mid.\mid\mid_{\mathrm{1}} \right)\:\mathrm{are}\:\mathrm{normed}\:\mathrm{space}. \\ $$$$\:\:\left(\mathrm{Helpe}\:\mathrm{me}\:\mathrm{please}\right) \\ $$

Question Number 120519 Answers: 1 Comments: 1