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Question Number 213007 Answers: 1 Comments: 0

can′t find coefficient f^((n)) (α) of Y_ν (z) formal power series of Y_ν (z) is Y_ν (z)=Σ_(h=0) ^∞ ((Y_ν ^((h)) (α))/(h!))(z−α)^h But.. can′t generalize coeff Y_ν ^((h)) (α) series representation ↓ Y_ν (z)=−(1/π)((2/z))^ν ∙Σ_(h=0) ^(ν−1) ((𝚪(ν−h))/(h!))((z/2))^(2h) +(2/π)J_ν (z)ln((1/2)z)−(1/π)((z/2))^ν ∙Σ_(h=0) ^∞ (((−1)^h (ψ^((0)) (h+1)−ψ^((0)) (h+ν+1)))/(h!(h+ν)!))((z/2))^(2h)

$$\mathrm{can}'\mathrm{t}\:\mathrm{find}\:\:\mathrm{coefficient}\:{f}^{\left({n}\right)} \left(\alpha\right)\:\mathrm{of}\:{Y}_{\nu} \left({z}\right) \\ $$$$\mathrm{formal}\:\mathrm{power}\:\mathrm{series}\:\mathrm{of}\:{Y}_{\nu} \left({z}\right)\:\mathrm{is} \\ $$$${Y}_{\nu} \left({z}\right)=\underset{{h}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{{Y}_{\nu} ^{\left({h}\right)} \left(\alpha\right)}{{h}!}\left({z}−\alpha\right)^{{h}} \\ $$$${But}..\:\mathrm{can}'\mathrm{t}\:\mathrm{generalize}\:\mathrm{coeff}\:{Y}_{\nu} ^{\left({h}\right)} \left(\alpha\right) \\ $$$$\mathrm{series}\:\mathrm{representation}\:\downarrow \\ $$$${Y}_{\nu} \left({z}\right)=−\frac{\mathrm{1}}{\pi}\left(\frac{\mathrm{2}}{{z}}\right)^{\nu} \centerdot\underset{{h}=\mathrm{0}} {\overset{\nu−\mathrm{1}} {\sum}}\:\frac{\boldsymbol{\Gamma}\left(\nu−{h}\right)}{{h}!}\left(\frac{{z}}{\mathrm{2}}\right)^{\mathrm{2}{h}} +\frac{\mathrm{2}}{\pi}{J}_{\nu} \left({z}\right)\mathrm{ln}\left(\frac{\mathrm{1}}{\mathrm{2}}{z}\right)−\frac{\mathrm{1}}{\pi}\left(\frac{{z}}{\mathrm{2}}\right)^{\nu} \centerdot\underset{{h}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{h}} \left(\psi^{\left(\mathrm{0}\right)} \left({h}+\mathrm{1}\right)−\psi^{\left(\mathrm{0}\right)} \left({h}+\nu+\mathrm{1}\right)\right)}{{h}!\left({h}+\nu\right)!}\left(\frac{{z}}{\mathrm{2}}\right)^{\mathrm{2}{h}} \\ $$

Question Number 213003 Answers: 1 Comments: 0

∫_0 ^1 (arctan x)^2 dx.

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{arctan}\:{x}\right)^{\mathrm{2}} {dx}. \\ $$$$ \\ $$

Question Number 213002 Answers: 1 Comments: 0

(((52+6(√(43)) )^(3/2) −(52−6(√(43)))^(3/2) )/(18))=?

$$\:\:\frac{\left(\mathrm{52}+\mathrm{6}\sqrt{\mathrm{43}}\:\right)^{\mathrm{3}/\mathrm{2}} −\left(\mathrm{52}−\mathrm{6}\sqrt{\mathrm{43}}\right)^{\mathrm{3}/\mathrm{2}} }{\mathrm{18}}=? \\ $$

Question Number 213001 Answers: 1 Comments: 1

⋖ 14^3 +15^3 +16^3 +...+24^3 +25^3

$$\:\:\:\:\:\:\underline{\underbrace{\lessdot}\cancel{} } \mathrm{14}^{\mathrm{3}} +\mathrm{15}^{\mathrm{3}} +\mathrm{16}^{\mathrm{3}} +...+\mathrm{24}^{\mathrm{3}} +\mathrm{25}^{\mathrm{3}} \\ $$

Question Number 213000 Answers: 2 Comments: 0

f(((x−3)/(x+1))) + f(((x+3)/(1−x))) = x , x≠ ± 1 f(x)=?

$$\:\:\:\mathrm{f}\left(\frac{\mathrm{x}−\mathrm{3}}{\mathrm{x}+\mathrm{1}}\right)\:+\:\mathrm{f}\left(\frac{\mathrm{x}+\mathrm{3}}{\mathrm{1}−\mathrm{x}}\right)\:=\:\mathrm{x}\:,\:\mathrm{x}\neq\:\pm\:\mathrm{1} \\ $$$$\:\:\:\mathrm{f}\left(\mathrm{x}\right)=? \\ $$

Question Number 212999 Answers: 1 Comments: 0

Find the number of non zero integer solution (x,y) to the equation ((15)/(x^2 y)) + (3/(xy)) − (2/x) = 2

$$\:\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{non}\:\mathrm{zero}\:\mathrm{integer}\: \\ $$$$\:\mathrm{solution}\:\left(\mathrm{x},\mathrm{y}\right)\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\:\:\:\:\frac{\mathrm{15}}{\mathrm{x}^{\mathrm{2}} \mathrm{y}}\:+\:\frac{\mathrm{3}}{\mathrm{xy}}\:−\:\frac{\mathrm{2}}{\mathrm{x}}\:=\:\mathrm{2}\: \\ $$

Question Number 212997 Answers: 0 Comments: 3

Question Number 212993 Answers: 0 Comments: 1

Question Number 212992 Answers: 1 Comments: 0

a>0. b>0. a+b=2. Find the minimum value of a^(√a) b^(√b) .

$${a}>\mathrm{0}.\:{b}>\mathrm{0}.\:{a}+{b}=\mathrm{2}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:{a}^{\sqrt{{a}}} {b}^{\sqrt{{b}}} . \\ $$

Question Number 212991 Answers: 1 Comments: 0

⇃

$$\:\:\:\:\:\underbrace{\downharpoonleft\underline{}\:} \\ $$

Question Number 212984 Answers: 0 Comments: 0

Notation : Soit A une partie de R. On appelle indicatrice de A, note^ e χ_A , l′application x { ((1 si x ∈ A)),((0 sinon)) :}. 1. Pour k dans N^∗ notons f_k : x (cos x)^(2k) . Montrer que (f_k )_(k∈N^∗ ) converge vers χ_(πZ) . 2. Soit n un parame^ tre fixe^ dans N. Pour k dans N^∗ notons g_k : x f_k (n!πx). Montrer que (g_k )_(k∈N^∗ ) converge vers une application g^((n)) a^ de^ terminer, de^ pendante du parame^ tre n. E^ crire g^((n)) sous la forme χ_A ou^ A est une partie de R a^ de^ terminer. 3. Montrer que la suite de fonctions (g^((n)) )_(n∈N) converge vers χ_Q . 4. Soit x dans R. Calculer lim_(n→∞) (lim_(k→∞) (cos(n!πx))^(2k) ).

$$\underline{\boldsymbol{\mathrm{Notation}}\::}\:\mathrm{Soit}\:{A}\:\mathrm{une}\:\mathrm{partie}\:\mathrm{de}\:\mathbb{R}.\:\mathrm{On}\:\mathrm{appelle}\:{indicatrice}\:{de}\:{A}, \\ $$$$\mathrm{not}\acute {\mathrm{e}e}\:\chi_{{A}} ,\:\mathrm{l}'\mathrm{application}\:{x}\: \:\begin{cases}{\mathrm{1}\:\mathrm{si}\:{x}\:\in\:{A}}\\{\mathrm{0}\:\mathrm{sinon}}\end{cases}. \\ $$$$ \\ $$$$\mathrm{1}.\:\mathrm{Pour}\:{k}\:\mathrm{dans}\:\mathbb{N}^{\ast} \:\mathrm{notons}\:{f}_{{k}} \::\:{x}\: \:\left(\mathrm{cos}\:{x}\right)^{\mathrm{2}{k}} . \\ $$$$\mathrm{Montrer}\:\mathrm{que}\:\left({f}_{{k}} \right)_{{k}\in\mathbb{N}^{\ast} } \:\mathrm{converge}\:\mathrm{vers}\:\chi_{\pi\mathbb{Z}} . \\ $$$$ \\ $$$$\mathrm{2}.\:\mathrm{Soit}\:{n}\:\mathrm{un}\:\mathrm{param}\grave {\mathrm{e}tre}\:\mathrm{fix}\acute {\mathrm{e}}\:\mathrm{dans}\:\mathbb{N}. \\ $$$$\mathrm{Pour}\:{k}\:\mathrm{dans}\:\mathbb{N}^{\ast} \:\mathrm{notons}\:{g}_{{k}} \::\:{x}\: \:{f}_{{k}} \left({n}!\pi{x}\right). \\ $$$$\mathrm{Montrer}\:\mathrm{que}\:\left({g}_{{k}} \right)_{{k}\in\mathbb{N}^{\ast} } \:\mathrm{converge}\:\mathrm{vers}\:\mathrm{une}\:\mathrm{application}\:{g}^{\left({n}\right)} \:\grave {\mathrm{a}} \\ $$$$\mathrm{d}\acute {\mathrm{e}terminer},\:\mathrm{d}\acute {\mathrm{e}pendante}\:\mathrm{du}\:\mathrm{param}\grave {\mathrm{e}tre}\:{n}.\:\acute {\mathrm{E}crire}\:{g}^{\left({n}\right)} \:\mathrm{sous} \\ $$$$\mathrm{la}\:\mathrm{forme}\:\chi_{{A}} \:\mathrm{o}\grave {\mathrm{u}}\:{A}\:\mathrm{est}\:\mathrm{une}\:\mathrm{partie}\:\mathrm{de}\:\mathbb{R}\:\grave {\mathrm{a}}\:\mathrm{d}\acute {\mathrm{e}terminer}. \\ $$$$ \\ $$$$\mathrm{3}.\:\mathrm{Montrer}\:\mathrm{que}\:\mathrm{la}\:\mathrm{suite}\:\mathrm{de}\:\mathrm{fonctions}\:\left({g}^{\left({n}\right)} \right)_{{n}\in\mathbb{N}} \:\mathrm{converge}\:\mathrm{vers}\:\chi_{\mathbb{Q}} . \\ $$$$ \\ $$$$\mathrm{4}.\:\mathrm{Soit}\:{x}\:\mathrm{dans}\:\mathbb{R}.\:\mathrm{Calculer}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\underset{{k}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{cos}\left({n}!\pi{x}\right)\right)^{\mathrm{2}{k}} \:\right). \\ $$

Question Number 212974 Answers: 1 Comments: 0

Question Number 212976 Answers: 2 Comments: 0

4^x = 125 and 8^y = 5 find: ((2x − y)/y) = ?

$$\mathrm{4}^{\boldsymbol{\mathrm{x}}} \:=\:\mathrm{125}\:\:\:\mathrm{and}\:\:\:\mathrm{8}^{\boldsymbol{\mathrm{y}}} \:=\:\mathrm{5} \\ $$$$\mathrm{find}:\:\:\:\frac{\mathrm{2x}\:−\:\mathrm{y}}{\mathrm{y}}\:=\:? \\ $$

Question Number 212986 Answers: 1 Comments: 1

Question Number 212950 Answers: 0 Comments: 0

Question Number 212949 Answers: 5 Comments: 0

Question Number 212982 Answers: 2 Comments: 0

Question Number 212936 Answers: 0 Comments: 1

Question Number 212935 Answers: 1 Comments: 0

Proving : ∣∫_0 ^1 f(x)dx−((f(0)+f(1))/2)∣≤(1/(32))

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Proving}\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mid\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}−\frac{{f}\left(\mathrm{0}\right)+{f}\left(\mathrm{1}\right)}{\mathrm{2}}\mid\leqslant\frac{\mathrm{1}}{\mathrm{32}} \\ $$$$ \\ $$

Question Number 212929 Answers: 1 Comments: 1

what is the difference between moment and torque?

$${what}\:{is}\:{the}\:{difference}\:{between}\:{moment} \\ $$$${and}\:{torque}? \\ $$

Question Number 212928 Answers: 2 Comments: 1

Question Number 212927 Answers: 1 Comments: 0

White horse≠horse and horse≠White horse Q:Are the above propositions equivalent?

$$\mathrm{White}\:\mathrm{horse}\neq\mathrm{horse}\:\mathrm{and}\:\mathrm{horse}\neq\mathrm{White}\:\mathrm{horse} \\ $$$${Q}:\mathrm{Are}\:\mathrm{the}\:\mathrm{above}\:\mathrm{propositions}\:\mathrm{equivalent}? \\ $$

Question Number 212926 Answers: 2 Comments: 0

find integers x,y such that (x/(x−3)) −(4/(y^2 −45)) = (1/(100))

$$\:\mathrm{find}\:\mathrm{integers}\:\mathrm{x},\mathrm{y}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\:\:\frac{\mathrm{x}}{\mathrm{x}−\mathrm{3}}\:−\frac{\mathrm{4}}{\mathrm{y}^{\mathrm{2}} −\mathrm{45}}\:=\:\frac{\mathrm{1}}{\mathrm{100}} \\ $$

Question Number 212921 Answers: 0 Comments: 7

Dear Tinkutara last days I face a problem: Very often I loose all my work when saving (“save file”). I get one or two blank lines which are saved in the relevant file. I use v.2.286, android 12, api level 31. Any suggestion or update would be grateful. Thank you.

$${Dear}\:{Tinkutara} \\ $$$${last}\:{days}\:{I}\:{face}\:{a}\:{problem}: \\ $$$${Very}\:{often}\:{I}\:{loose}\:{all}\:{my}\:{work}\:{when} \\ $$$${saving}\:\left(``{save}\:{file}''\right).\:{I}\:{get}\:{one}\:{or} \\ $$$${two}\:{blank}\:{lines}\:{which}\:{are}\:{saved}\:{in} \\ $$$${the}\:{relevant}\:{file}. \\ $$$${I}\:{use}\:{v}.\mathrm{2}.\mathrm{286},\:{android}\:\mathrm{12},\:{api}\:{level}\:\mathrm{31}. \\ $$$${Any}\:{suggestion}\:{or}\:{update}\:{would}\:{be} \\ $$$${grateful}. \\ $$$${Thank}\:{you}. \\ $$

Question Number 212901 Answers: 1 Comments: 2

Question Number 212896 Answers: 0 Comments: 2

∫(dx/(x^5 −1))

$$\int\frac{{dx}}{{x}^{\mathrm{5}} −\mathrm{1}} \\ $$

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