Let f(z) = ((z -3i)² + 9)ez-31 The Laurent series representation of f(z) in the domain 0 < |z-3i| <∞o. a) (z − 3i)² + (z − 3i) + Σn=0((n+2)! + i)(z-31)² 1 1 b) 2(z-3i) + En=0; n! (z-31)n c) 9 + 9(z-3i) + Σn=2((n-2)! 2 ((1²2): + 2²/1) (2 − 3i)* O a. O b. O C.
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- A periodic function is defined as follows: (3 for-1 ≤ t < 0 16 for 0 ≤t<1 f(t) = {2 where f (t + 2) = f(t). Find the first four non-zero coefficients for the Fourier series expansion.if the Fourier series coefficients for the signal X[n] with a period N are "ax", calculate the Fourier series coefficients of the following expressions in terms of "ax". (a) x[n-2]+ x[n-(N-4)/2] (N even) (b) -1"x[-n] (N even) (c) yin]={ x[n], it n even 0, otherwise2)Given oDetermine the Fourier Series of: f(t) = { { Α= A f® = ΣΑ#t-1)*+1 + B]D(nat) f(t) n=C Β = C = -3, 3, D = -12x 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 0Q) find Fourier series on [-7,1] – 1 |4 -πIf f(r) = E , g(x) = r'f(5x), then the series representation of g(1) is 3D (-1)* 5 +1 B) E, (-1)* k 5 k C) Σ (-1) k 54 +1 D ) Σ. E) E, -D k5 +1 .A .B .C .DO .E I I 1 I13) The integer n = a) If n = 3 (mod 9), then what is x ? b) If n = 3 (mod 11), then what is x ? Hint for #13: 12700140x3243243 is missing the digit x. Consider the number 1234. By writing each digit in scientific notation, we have 1234 1 x 10³ +2× 10² +3 × 10¹ + 4 × 10⁰ Then notice that 10 = 1 (mod 9). When we reduce mod 9, we obtain 1234 1 x 10³ +2× 10² +3 x 10¹ + 4 x 10⁰ = = 1x 1³ + 2 x 1² + 3 × 1¹ + 4 × 1⁰ = 1+ 2+ 3+ 4 = 10 = 1 (mod 9). Since 10 = -1 (mod 11), when we reduce mod 11, we obtain 1234 1 x 10³ +2× 10² +3 x 10¹ + 4 x 10⁰ = = 1x (-1)³ +2× (-1)² +3 × (-1)¹ +4× (-1)⁰ = -1 + 2-3 + 4 = 2 (mod 11).The Taylor expansion of f(z) = ze²*4 around z = -i is -ie?- + (z +1)* (n-1) O equal to the above O None of these +)e²-" (z + t)" e(z+ i)" (n-1)! equal to the above equal to the above1)Determine S[f] (Fourier series) if: d) f(x)=ex+x ,x∈ [-1, 1] such that f(x) = f(x + 2)Find the Taylor's or Laurent's series expansion of the complex variable function which is represented by : f(z) = ; i) 1< ]z] < 2 ii) ]z|< 2. Also (z²–1)(z²+4) z2 classify the singularity of f(z) %3D (z-2)ez-1Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,