Let n e N. You are going to prove that 1.3-5.. (2n – 1) = 27 2 -4 ·6 . .· (2n) 27 In := | cos2" 0 d0 (a) Relate In to the following integral (++) 2n Jn := {lz|=1}
Let n e N. You are going to prove that 1.3-5.. (2n – 1) = 27 2 -4 ·6 . .· (2n) 27 In := | cos2" 0 d0 (a) Relate In to the following integral (++) 2n Jn := {lz|=1}
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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
Transcribed Image Text:Let n e N. You are going to prove that
1.3-5.. (2n – 1)
= 27
2 -4 ·6 . .· (2n)
27
In :=
| cos?" 0 do
(a) Relate In to the following integral
2n
:=
{l2|=1}
1
(b) For an appropriate zo € C and an appropriate analytic function f(z), write Jn in the
form
{(2)
dz .
(c) Using the Cauchy Integral Formula, show that
f(2)
2in
g(2n) (z0) .
(2n)!
dz =
=|=1}
I+uz(0z – 2)
(d) Put together the above pieces to obtain the value of In.
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