Question 5 1.Supply all the details of the proof of Gauss's Mean Value Theorem:If f is analytic in a simply connected domain D that contains the circle CR ,2лcentered at zo with radius R, then f(zo) =-1+ Re" )d0.2.Let a function ƒ be continuous on a closed bounded region R. and let it be analyticand not constant throughout the interior of R. Assuming that f(z) ± 0 anywhere inR, prove that [(z)| has a minimum value m in R which occurs on the boundary of Rand never in the interior of R by applying the Maximum Modulus Principle to thefunction 1/f(z).3.Find the sum of the series.001+i8.1(a)(b)Σ2n=0n=2 (2+i)"eiz- eiz4.Use the Maclaurin series e?Σand the definition sin(z)=%Dn=0 n!to find the Maclaurin series for the entire function f(z) = sin(z).2iDifferentiate the Maclaurin series for f(z) = sin(z) term by term to obtain theMaclaurin series for f (z) = cos(z).5.6.Derive the Taylor series representation in the disk |z – i| < /2 :(z-i)"Σ1-z 0 (1-i)"+!11Hint: Start by writing1-z11-i 1-(z-i)/(1– i)1%3D%3D(1- i)- (z-i)203

Name: Question 5 1.Supply all the details of the proof of Gauss's Mean Value
Uploaded: 2024-03-20T07:07:50.000Z
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Description: Question 5 1.Supply all the details of the proof of Gauss's Mean Value Theorem:If f is analytic in a simply connected domain D that contains the circle CR ,2лcentered at zo with radius R, then f(zo) =-1+ Re" )d0.2.Let a function ƒ be continuous on a