Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 74914 by arkanmath7@gmail.com last updated on 03/Dec/19

$$Cansomeonesolvethisquestionplz?$$$$solvethecontourintegral$$$${\int }_C\frac{e^{iz}}{z^3}dzwhereCisthecircle\mid z\mid =2$$

Answered by mind is power last updated on 05/Dec/19

$$\mathrm{pols}\mathrm{of}\mathrm{f}\left(\mathrm{z}\right)=\frac{{\mathrm{e}}^{\mathrm{iz}}}{{\mathrm{z}}^3}\mathrm{has}\mathrm{pol}\mathrm{at}\mathrm{z}=0\mathrm{order}3$$$$\int \frac{{\mathrm{e}}^{\mathrm{iz}}}{{\mathrm{z}}^3}\mathrm{dz}=2\mathrm{i}\pi \mathrm{Res}(\mathrm{f},0)=2\mathrm{i}\pi .\frac{1}{2!}\frac{{\mathrm{d}}^2}{{\mathrm{dz}}^2}.{\mathrm{z}}^3\mathrm{f}\left(\mathrm{z}\right){\mid }_{\mathrm{z}=0}$$$$=\frac{1}{2}2\mathrm{i}\pi (-{\mathrm{e}}^{\mathrm{iz}}){\mid }_{\mathrm{z}=0}=-\mathrm{i}\pi$$

Commented by arkanmath7@gmail.com last updated on 04/Dec/19

$$thisqust.{isn}^{\prime }tsolvedfrom{Schoum}^{\prime }sandthe$$$$answeris-\pi i$$

Commented by mind is power last updated on 05/Dec/19

$$\mathrm{yeah}\mathrm{Res}(\mathrm{f},0)=\frac{1}{2!}\frac{{\mathrm{d}}^2}{{\mathrm{dz}}^2}{\mathrm{z}}^3\mathrm{f}\left(\mathrm{z}\right){\mid }_{\mathrm{z}=0}$$

Commented by arkanmath7@gmail.com last updated on 05/Dec/19

$$thankyousir$$

Commented by mind is power last updated on 05/Dec/19

$${\mathrm{y}}^{\prime }\mathrm{re}\mathrm{welcom}$$$$\mathrm{we}\mathrm{can}\mathrm{generaliz}\mathrm{it}$$$${\int }_{\mathrm{C}}\frac{{\mathrm{e}}^{\mathrm{iz}}}{{\mathrm{z}}^{\mathrm{n}}}\mathrm{dz}=\frac{2\mathrm{i}\pi }{(\mathrm{n}-1)!}.{\left(\mathrm{i}\right)}^{\mathrm{n}-1}$$$$$$$$=\frac{2\pi .{\left(\mathrm{i}\right)}^{\mathrm{n}}}{(\mathrm{n}-1)!}$$

Commented by arkanmath7@gmail.com last updated on 06/Dec/19

$$goodworkthanksagain$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com