let-C-z-z-1-calculste-C-tanz-dz-

Question Number 125505 by mathmax by abdo last updated on 11/Dec/20

let C = {z / ∣ z ∣= 1} calculste \int_{C} tanz dz

Answered by mathmax by abdo last updated on 11/Dec/20

\int_{C} tanz dx = \int_{C} \frac{sinz}{cosz} dz =_{e^{iz} = w} \int_{∣ w ∣= 1} \frac{\frac{w - w^{- 1}}{2 i}}{\frac{w + w^{- 1}}{2}} \frac{dw}{iw}

= \frac{1}{i} \int_{∣ w ∣= 1} \frac{w - w^{- 1}}{iw (w + w^{- 1})} dw = - \int_{∣ w ∣= 1} \frac{w - w^{- 1}}{w^{2} + 1} dw

= - \int_{∣ w ∣= 1} \frac{w^{2} - 1}{w (w^{2} + 1)} dw let φ (w) = \frac{w^{2} - 1}{w (w^{2} + 1)} the poles of w

are 0 = i and - i \Rightarrow \int_{∣ w ∣= 1} φ (w) dw = 2 i π {Res (φ, o) + Res (φ, i) + Res (φ, - i)}

Res (φ, o) = - 1 and Res (φ, i) = \frac{- 2}{i (2 i)} = 1

Res (φ, - i) = \frac{- 2}{- i (- 2 i)} = \frac{- 2}{- 2} = 1 \Rightarrow

\int_{C} tanz dz = - 2 i π {- 1 + 1 + 1} = - 2 i π