To show that P, (x) is the coefficient of z" in the expansion of (x-2xz +2²)in ascending powers of z,or,(1 - 2x2 +2²) 2-Σ z" P₁ (x),n=0|x|≤1₁ |z|<1

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Answered: To show that P, (x) is the coefficient…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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To show that P, (x) is the coefficient of " in the expansion of (1-2xz+2 Z in ascending powers of z, or, 8 (1-2x² +2²) ³ - [ 2² 1 n=0 Σ z" P₁ (x), |x|≤ 1, \2\<1
Suppose that f(r) is a function given by for 0<1< T, f(z) = for -n
The Taylor expansion of f (z) = ze²+2 around z = -i is None of these E -De2-i (z+ i)" (n-1)! equal to the above -ie?-i + equal to the above +)e2-i (z + i)" (n-1)! n! equal to the above
9. Let f(r) = x[a.b) (x) be the characteristic function of the interval [a, b] c [-7, 7], that is, X[a.6) (2x) = { ! if x € [a, b], 0 otherwise. (a) Show that the Fourier series of ƒ is given by e-ina – e-inb „inz b - a f(r) ~ 27 2тin n#0 The sum extends over all positive and negative integers excluding 0. (b) Show that if a + –n or b# a and a + b, then the Fourier series does not converge absolutely for any r. [Hint: It suffices to prove that for many values of n one has | sin n6o| 2c> 0 where 09 = (b – a)/2.]
5. What is the coefficient of x''y" in the expansion (– 3x +4y)ª. Show your work here:
Use the following information to answer the next question Jacob travels from home to school on his motor scooter. He can only travel on the roads indicated on the diagram. School Home 14. The number of different routes Jacob can take without backtracking is 6435 146 O 1451520 2520
Let f:[-T, 1]→ R, f(x) = x then the trigonometric Fourier series of is given by (-1)" cos(nx) and it +4 n =1 n? converges O uniformly on [-n,n]. O converges in the mean, i.e. lim f - sall = 0, where s, is the nth partial sum of the Fourier series of f. n+ 00 O converges pointwise but not uniformly on (- T, TT). converges in the mean (i.e. lim f – sll=0, where sn is the nth partial sum of the Fourier series of f.), but %3D not pointwise. converges in the mean (i.e. lim f-Sn = 0, where s, is the nth partial sum of the Fourier series of f)and n- 00 uniformly on [-T, T]. O uniformly on (-1, TT) and converges to 0 at xo = + TT.

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To show that P, (x) is the coefficient of " in the expansion of (x-2xz+z² Z in ascending powers of z, or, (1 − 2xz + 2²)¯ ½ = Σ z″ P‚, (x), n=0 x≤1, 2<1