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Unformatted text preview: Calculus IVA Complex variables. Lecture II November 1, 2003 1 Complex line integrals Consider an oriented curve C in the plane. We have defined the line integral Z C Pdx + Qdy . If x = x ( t ) , y = y ( t ) , a ≤ t ≤ b is a parametrization of the curve, then Z C Pdx + Qdy = Z b a [ P ( x ( t ) , y ( t )) x ( t ) + Q ( x ( t ) , y ( t )) y ( t )] dt . We define similarly a complex line integral Z C f ( z ) dz where f ( z ) is a complex valued function. We write z = x + iy dz = dx + idy 1 f = P + iQ where P, Q are the real and imaginary parts of f . Then Z C f ( z ) dz = Z C ( P + iQ )( dx + idy ) = Z C Pdx Qdy + i Z C Qdx + Pdy . This can computed directly from the parametrization: Z C f ( z ) dz = Z C f ( z ( t )) dz dt dt since dz dt = dx dt + i dy dt . Example 1 : compute Z C 1 z dz where C is the circle of radius a and center 0 with counterclockwise orienta tion. Solution We parametrize the circle by x = a cos( θ ) , = a sin( θ ) , ≤ θ ≤ 2 π , or, in complex form: z = ae iθ . Thus dz = iae iθ dθ . Hence Z C 1 z dz = Z 2 π iae iθ ae iθ dθ = i Z 2 π dθ = 2 iπ . Example 2 : compute Z C zdz where C is as above. Solution Again we find Z C zdz = Z 2 π ae iθ iae iθ dθ = ia 2 Z 2 π e 2 iθ dθ = ia 2 2 1 e iθ  θ =2 π θ =0 = 0 . 2 2 The fundamental theorem of calculus for holomorphic functions Theorem 1 Suppose that f is holomorphic in a connected domain D and suppose that f has an holomorphic antiderivative F in D . Let A and B be two points of D . Then for any oriented curve C from A to B contained in D we have Z C f ( z ) dz = F ( B ) F ( A ) . In particular, if A = B (that is, if C is closed), then Z C f ( z ) dz = 0 . We parametrize the curve: z = z ( t ) , a ≤ t ≤ b , z ( a ) = A , z ( b ) = B ....
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 Spring '11
 Peters
 Calculus, Derivative, Integrals, dz, dθ, dζ

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