complex-analysis-C-2z-sinz-2-z-2-z-3-dz-C-Z-5-

Question Number 129688 by Engr_Jidda last updated on 17/Jan/21

c o m p l e x a n a l y s i s

\oint_{C} \frac{ϱ^{2 z} + {s i n z}^{2}}{(z - 2) (z - 3)} d z C :∣ Z ∣= 5

Answered by mathmax by abdo last updated on 17/Jan/21

φ (z) = \frac{e^{2 z} + \sin (z^{2})}{(z - 2) (z - 3)} the poles of φ are 2 and 3

\int_{∣ z ∣= 5} φ (z) dz = 2 i π {Res (φ, 2) + Res (φ, 3)}

= 2 i π {\frac{e^{4} + \sin (4)}{- 1} + \frac{e^{6} + \sin 9}{1}} = 2 i π {e^{6} - e^{4} + \sin 9 - \sin 4}