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Question Number 55070 Answers: 0 Comments: 1

Factorised the polynom z^4 +1 be polynom with lower degree, but have real coefficient

$$\mathrm{Factorised}\:\mathrm{the}\:\mathrm{polynom}\:{z}^{\mathrm{4}} +\mathrm{1}\: \\ $$$$\mathrm{be}\:\mathrm{polynom}\:\mathrm{with}\:\mathrm{lower}\:\mathrm{degree}, \\ $$$$\mathrm{but}\:\mathrm{have}\:\mathrm{real}\:\mathrm{coefficient} \\ $$

Question Number 55069 Answers: 1 Comments: 3

Known analytic function f(z)=((2(z−2))/(z(z−4))) and written as f(z)=Σ_(n=0) ^(∝) a_n (z−1)^n The value of a_(100) is...

$$\mathrm{Known}\:\mathrm{analytic}\:\mathrm{function} \\ $$$${f}\left({z}\right)=\frac{\mathrm{2}\left({z}−\mathrm{2}\right)}{{z}\left({z}−\mathrm{4}\right)} \\ $$$$\mathrm{and}\:\mathrm{written}\:\mathrm{as}\:{f}\left({z}\right)=\underset{{n}=\mathrm{0}} {\overset{\propto} {\Sigma}}\:{a}_{{n}} \left({z}−\mathrm{1}\right)^{{n}} \\ $$$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:{a}_{\mathrm{100}} \:\mathrm{is}... \\ $$

Question Number 55068 Answers: 0 Comments: 0

Calculate value of ∫_C e^(2/z) dz if C is unit circle

$$\mathrm{Calculate}\:\mathrm{value}\:\mathrm{of}\:\int_{{C}} {e}^{\frac{\mathrm{2}}{{z}}} \:{dz} \\ $$$$\mathrm{if}\:{C}\:\mathrm{is}\:\mathrm{unit}\:\mathrm{circle} \\ $$$$ \\ $$

Question Number 55067 Answers: 0 Comments: 0

Known C circle centered at 0. Find value than ∫_C (dz/(1−z))

$$\mathrm{Known}\:\:{C}\:\mathrm{circle}\:\mathrm{centered}\:\mathrm{at}\:\mathrm{0}. \\ $$$$\mathrm{Find}\:\mathrm{value}\:\mathrm{than}\:\int_{{C}} \frac{{dz}}{\mathrm{1}−{z}} \\ $$

Question Number 55066 Answers: 0 Comments: 0

Find radius convergence for series 1−z^2 +z^4 −z^6 +...

$$\mathrm{Find}\:\mathrm{radius}\:\mathrm{convergence}\:\mathrm{for}\:\mathrm{series} \\ $$$$\mathrm{1}−{z}^{\mathrm{2}} +{z}^{\mathrm{4}} −{z}^{\mathrm{6}} +... \\ $$

Question Number 55059 Answers: 0 Comments: 0

Find the number of root equation z^4 −5z+1=0 in 1 ≤ ∣z∣ ≤ 2

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{root}\:\mathrm{equation} \\ $$$${z}^{\mathrm{4}} −\mathrm{5}{z}+\mathrm{1}=\mathrm{0}\:\mathrm{in}\:\mathrm{1}\:\leqslant\:\mid{z}\mid\:\leqslant\:\mathrm{2} \\ $$

Question Number 55058 Answers: 0 Comments: 0

Prove that: P_n (z)=a_0 z^n +a_1 z^(n−1) +...+a_(n−1) z+a_n at least have one value of zero

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{P}_{{n}} \left({z}\right)={a}_{\mathrm{0}} {z}^{{n}} +{a}_{\mathrm{1}} {z}^{{n}−\mathrm{1}} +...+{a}_{{n}−\mathrm{1}} {z}+{a}_{{n}} \\ $$$$\mathrm{at}\:\mathrm{least}\:\mathrm{have}\:\mathrm{one}\:\:\mathrm{value}\:\mathrm{of}\:\mathrm{zero} \\ $$

Question Number 55057 Answers: 0 Comments: 0

Prove that Ln(z+1)=z−(z^2 /2)+(z^3 /3)−(z^4 /4)+... for ∣z∣< 1

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{Ln}\left({z}+\mathrm{1}\right)={z}−\frac{{z}^{\mathrm{2}} }{\mathrm{2}}+\frac{{z}^{\mathrm{3}} }{\mathrm{3}}−\frac{{z}^{\mathrm{4}} }{\mathrm{4}}+... \\ $$$$\mathrm{for}\:\mid{z}\mid<\:\mathrm{1} \\ $$

Question Number 55055 Answers: 1 Comments: 0

Question Number 55054 Answers: 0 Comments: 0

Question Number 55052 Answers: 0 Comments: 3

Question Number 55051 Answers: 0 Comments: 1

calculate lim_(n→+∞) ∫_0 ^1 (dx/(1+x+x^2 +...+x^n ))

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{\mathrm{1}+{x}+{x}^{\mathrm{2}} \:+...+{x}^{{n}} } \\ $$

Question Number 55454 Answers: 0 Comments: 3

let f(a) =∫_0 ^∞ ((ln(x))/(x^2 +a)) with a>0 1) calculate f(a) intermsof a 2) find the values of ∫_0 ^∞ ((ln(x))/(x^2 +1))dx and ∫_0 ^∞ ((ln(x))/(x^2 +2))dx 3) let g(a) =∫_0 ^∞ ((ln(x))/((x^2 +a)^n )) dx .calculate g(a) interms of a 4) find values> of ∫_0 ^∞ ((ln(x))/((x^2 +3)^2 ))dx 5) find nature of the serie Σ f(n) andΣ g(n)

$${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left({x}\right)}{{x}^{\mathrm{2}} \:+{a}}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right)\:{intermsof}\:{a} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({x}\right)}{{x}^{\mathrm{2}} +\mathrm{1}}{dx}\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left({x}\right)}{{x}^{\mathrm{2}} \:+\mathrm{2}}{dx} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{g}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left({x}\right)}{\left({x}^{\mathrm{2}} \:+{a}\right)^{{n}} }\:{dx}\:\:.{calculate}\:{g}\left({a}\right)\:{interms}\:{of}\:{a} \\ $$$$\left.\mathrm{4}\right)\:{find}\:{values}>\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left({x}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{5}\right)\:{find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:{f}\left({n}\right)\:\:{and}\Sigma\:{g}\left({n}\right)\: \\ $$

Question Number 55044 Answers: 1 Comments: 2