written 6.1 years ago by | • modified 2.2 years ago |
Find the values of:
(i) φ(1)
(ii) φ' (2)
(iii) φ(3)
Subject: Applied Mathematics 4
Topic: Complex Integration
Difficulty: Medium
written 6.1 years ago by | • modified 2.2 years ago |
Find the values of:
(i) φ(1)
(ii) φ' (2)
(iii) φ(3)
Subject: Applied Mathematics 4
Topic: Complex Integration
Difficulty: Medium
written 5.9 years ago by |
C = (0,2); r = 3
$\phi(\alpha) = 2 \pi i[\alpha e^\alpha]$ is $\alpha$ lies inside
= 0 if $\alpha$ lies outside
(i) z = 1 = (1,0)
Let A = (1,0), C = (0,2)
d(AC) = $ \sqrt{(1-0)^2 + (0-2)^2} = \sqrt{1+4} = \sqrt{5} \lt 3 $
Therefore, A = (1,0) lies outside.
$ \therefore \phi(1) = 2 \pi i[1e^1] = 2 \pi i e $
(ii) z = 2 = (2,0)
Let, B = (2,0); C = (0,2)
d(BD) = $ \sqrt{2^2 + 2^2} = \sqrt{4+4} = \sqrt{8} \lt 3 $
Therefore, B lies inside.
$ \phi'(\alpha) = 2 \pi i [\alpha e^\alpha + e^\alpha] = 2 \pi i [2e^2 + e^2] \\ = 2 \pi i (e^2)(2+1) = 2 \pi i(e^2)(3) = 6 \pi i e^2 $
(iii) z = 3 = (3,0)
D = (3,0); C = (0,2)
d(CD) = $ \sqrt{(3-0)^2 + (0-2)^2} = \sqrt{3^2 + (-2)^2} + \sqrt{9+4} = \sqrt{13} \gt 3 $
Therefore D lies outside.
$ \therefore \phi(3) = 0 $