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Complex Integration — Howell & Matthews
Mathematics — Complex Analysis · contour-integrals-branch
Problem Parametrize C 1 + ( 0 ) C_1^+(0) C 1 + ( 0 ) z ( t ) = cos t + i sin t z(t) = \cos t + i\sin t z ( t ) = cos t + i sin t − π ≤ t ≤ π -\pi \leq t \leq \pi − π ≤ t ≤ π z 1 2 = r 1 2 cos θ 2 + i r 1 2 sin θ 2 z^{\frac{1}{2}} = r^{\frac{1}{2}}\cos\dfrac{\theta}{2} + ir^{\frac{1}{2}}\sin\dfrac{\theta}{2} z 2 1 = r 2 1 cos 2 θ + i r 2 1 sin 2 θ − π < θ ≤ π -\pi < \theta \leq \pi − π < θ ≤ π ∫ C 1 + ( 0 ) z 1 2 d z \displaystyle\int_{C_1^+(0)} z^{\frac{1}{2}}\,dz ∫ C 1 + ( 0 ) z 2 1 d z t → − π t \to -\pi t → − π Browse the problem library