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

Question Number 148371 by mathmax by abdo last updated on 27/Jul/21

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

Answered by mathmax by abdo last updated on 27/Jul/21

let f (z) = {ze}^{\frac{2}{z^{2}}} le seul point singulier de f is 0 \Rightarrow

\int_{γ} f (z) dz = 2 i π Res (f, o) on determine la serie de laurent

on e^{u} = \sum_{n = 0}^{\infty} \frac{u^{n}}{n!} \Rightarrow e^{\frac{2}{z^{2}}} = \sum_{n = 0}^{\infty} \frac{1}{n!} {(\frac{2}{z^{2}})}^{n}

= \sum_{n = 0}^{\infty} \frac{2^{n}}{n! z^{2 n}} \Rightarrow f (z) = z \sum_{n = 0}^{\infty} \frac{2^{n}}{n! z^{2 n}}

= z {1 + \frac{2}{1! z^{2}} + \frac{4}{2! z^{4}} + \dots} = z + \frac{2}{z} + \frac{4}{{2 z}^{3}} + \dots \Rightarrow Res (f, o) = 2 \Rightarrow

\int_{γ} f (z) dz = 2 i π (2) = 4 i π