Question 1 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 ,1 27centered at zo with radius R, then f(zo) =-+ Re" )d0.2л2.Let a function ƒ be continuous on a closed bounded region R, and let it be analytieand 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 ofR 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)"izz"Σand the definition sin(z)=eiz4.Use the Maclaurin series e?n=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 :1(z-i)"Σn=0 (1-i)"+11-z11Hint: Start by writing1-z1%3D%3D(1- i) - (z - i) 1-i 1-(z-i)/(1- i)203

Name: Question 1 1.Supply all the details of the proof of Gauss's Mean Value
Uploaded: 2024-03-20T07:07:50.000Z
Channel: Bartleby
Description: Question 1 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 ,1 27centered at zo with radius R, then f(zo) =-+ Re" )d0.2л2.Let a function ƒ be continuous on