calculate-z-3-e-1-z-2-dz-with-t-3e-it-and-t-0-2pi-

Question Number 148372 by mathmax by abdo last updated on 27/Jul/21

calculate \int_{γ} z^{3} e^{\frac{1}{z^{2}}} dz with γ (t) = {3 e}^{it} and t \in [0, 2 π]

Answered by mathmax by abdo last updated on 28/Jul/21

let f (z) = z^{3} e^{\frac{1}{z^{2}}} le seul point singulier de f est o

ona \int γ f (z) dz = 2 i π Res (f, o) ona

e^{\frac{1}{z^{2}}} = \sum_{n = 0}^{\infty} \frac{1}{n! z^{2 n}} \Rightarrow z^{3} e^{\frac{1}{z^{2}}} = z^{3} \sum_{n = 0}^{\infty} \frac{1}{n! z^{2 n}}

= z^{3} {1 + \frac{1}{z^{2}} + \frac{1}{{2 z}^{4}} + \frac{1}{{6 z}^{6}} + \dots} = z^{3} + z + \frac{1}{2 z} + \dots . \Rightarrow

Res (f, o) = \frac{1}{2} \Rightarrow \int_{γ} f (z) dz = 2 i π \times \frac{1}{2} = i π