1- Find the singular point of the function f(z) = and obtain the principle part of %3D z-sin(z) the Laurent series expansion of f(z). 2- Evaluate (CCW) z coshnz 2+13z2+36 dz, C: z| = n. 2 %3D
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- Find the Taylor series for f(x) = centered at a= 2. (-1)" +' (n+ 1) (A) E 00 (x- 2)" (B) E (-1)"+1 (n+ 1) 2n +2 (x – 2)" (C) E (-1)" (n + 1) 2n+1 (x – 2)" n=0 n=0 2" +2 n=0 00 (-1)" (n+ 1) 2" +1 00 (D) E (x- 2)" (E) E (-1)"+1 (x- 2)" (F) E -1)" 22 (x– 2)" (G) S. 2" (-1)"+1 n=0 2n +1 (x – 2)" n=0 on n=0 n=0 (H) E 2n+T (x – 2)" n=0 (-1)"Q 3. Find the Taylor series of f at the point c for : (c) f(x) = sin x, c= T (d) f(x) = cos x, c = 5 (a) f(x) = e", c= 2 (b) f(x) = }, c = 1f(t) {3t, 0<t<2} f(t){0, -2<t<0} f(t)= f(t+4) i) Sketch the graph of f (t) over the interval[−6, 4].ii) Find the Fourier series expansion off (t ) .
- 2)Given o2x 2, -3 < x < 2 Let f (x) = 4, 2 < x < 3 } be a periodic function with period 6. In such a case the value of the Fourier series of f at x = 2 is A) 2.5 B) 2 C) 4 D) 3.5 E) 3Determine the nth partial sum of the Fourier Series of: + x, - T < x < 0 f(x) = х, 2 0(3) The exponential of a number x can be computed through the Taylor series e=1+2+2/2++"/n!+... e² = 1 + x + x² / 2 + ··· + x^ /n! + ... The exponential of a square matrix A is defined by the Taylor series: A² A³ 2 3! An e₁ = I + A+ + + + +... where I is the identity matrix of the same size as A. n! (a) Find e for 0 2 3 B == 004 000 As you will learn, the solution of the system of first order differential equations d x= Ax dt with initial condition (0) Fois = x(t) = e¹¹ão, where et=I+ At+ (At)² (At)³ (At)" + + + + 2! 3! n! A square matrix A is said to be diagonal, if all of its entries outside of the main diagonal, (a), are equal to zero. If A is a diagonal matrix, then it is very easy to calculate e, For example, let Then eA - (61) + ( 0 -(32) (1+3+ (3)² + .. -(63) 0 A 1-(2) + ( 3 0 02 2! 0 (2)2 + 3! +. 2! 0 3! 0 1+2+(2)²+ -1 (b) What is e, where D = 0 40 ? 0 00Q) find Fourier series on [-7,1] – 1 |4 -π3z2-z+1 Q5: Find the Taylor series of f(z) = with center zo= 7. z2-9z+182- Find the taylor series: f(x) = cosz at Z₂ = I (find 5)(z)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.The Taylor series of the function f(x) = cosx at x=0 is (1) ²n En-0 n=0 (2n)! 2n=0 (2n)! Option 1 00 •2m=0 (n+1)! Option 3 (-1)+1 O Option 2 (−1)ngưn (-1) 2n+2 ·m=0 (2n+2)! O Option 4SEE MORE QUESTIONSRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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