Expand the followings in a Laurent's series f(2)= z-1 (i) about z=2 z2 -1 f(2)=7 when a) |z| <2 b) 2
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Q: Example Hw Consider laurent Series expansions around Zz. of the Bollowing o fiz)> 1-Z
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A: We need to find the Laurent series expansion of given function f(z) in the region 0<|z+1|<2.
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Q: 4 z + 3 2 Translate the function f(z)= (z)( z- 3)( z – 2) as Laurent series: (i) with in Iz| = 1…
A: We express in simpler form then find Laurent series.
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Q: Example Hw Consider laurent Series expansions around zzo of the following O f(2) = %3D 22-32+2
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Q: 8z +1 in a Laurent series valid for 0 < |z| < 1. (Ъ) Еxpand f(z) — z(1 – 2)
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Q: Show that for every decreasing sequence of events, {E,,n2 1}, lim P(E,)= P(lim E,)
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Q: Find the Laurent's series of f(z) = valid in the region (z-1)(z- 2)
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- Help7x2 + 3! 7x 7x4 The first derivative of the power series f(x) = 7+- - is 1! 2! 4! O f(x) O - f(x) O -7 f(x) O 7 f(r)m(m - 1) m(m – 1) . (m-k + 1) (1+ x)™ : =1+ mx+ x* +... 2! k! to obtain the MacLaurin series for (a) (1+) k=0 (b) V1+x = 1-(1/4)x+(5/32)x^(2)-(45/384)x^(3) +.... Enter first 4 terms only.
- find the series Laurent f(z) = Z-3 Z+2 121 >, < 121Find the function y1 of t which is the solution of 121y′′+88y′+12y=0 with initial conditions y1(0)=1,y′1(0)=0. Find the function y2 of t which is the solution of 121y′′+88y′+12y=0 with initial conditions y2(0)=0,y′2(0)=1. Find the Wronskian W(t)=W(y1,y2).2 (a) Page 3 of 11 (0) Module code: CAPE200001 (ii) (ii) Using the power series expansion for exp(-x) or otherwise, find the first 3 non-zero terms in the power series expansion of exp(-2x¹). Remember that exp(-x) is the same as e. Calculate lim Use the power series from (i) to find the limit of 1-2²-e 1-exp(-2x¹) Turn the page over using l'Hôpital's rule. 2²-exp(-2²) as x→0.What is the explicit function and recurrence function that defines the sequence f(1) = 2, f(2) = 6, f(3) =10, f(4) = 14, f(5) = 18?(1) A sequence an is given by a₁ = √2, an+1 = √2+ an. (a) By induction or otherwise, show that an is increasing and bounded above by 3. Apply the Monotonic Sequence Theorem to show that exists. (b) Find limn-+0 an.A sequence (n)-1 is said to oscillate if lim inf xnRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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