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Chapter4

# Chapter4 - Chapter 4 Integration Everybody knows that...

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Chapter 4 Integration Everybody knows that mathematics is about miracles, only mathematicians have a name for them: theorems. Roger Howe 4.1 Definition and Basic Properties At first sight, complex integration is not really anything different from real integration. For a continuous complex-valued function φ : [ a, b ] R C , we define Z b a φ ( t ) dt = Z b a Re φ ( t ) dt + i Z b a Im φ ( t ) dt . (4.1) For a function which takes complex numbers as arguments, we integrate over a curve γ (instead of a real interval). Suppose this curve is parametrized by γ ( t ) , a t b . If one meditates about the substitution rule for real integrals, the following definition, which is based on (4.1) should come as no surprise. Definition 4.1. Suppose γ is a smooth curve parametrized by γ ( t ) , a t b , and f is a complex function which is continuous on γ . Then we define the integral of f on γ as Z γ f = Z γ f ( z ) dz = Z b a f ( γ ( t )) γ 0 ( t ) dt . This definition can be naturally extended to piecewise smooth curves, that is, those curves γ whose parametrization γ ( t ), a t b , is only piecewise differentiable, say γ ( t ) is differentiable on the intervals [ a, c 1 ] , [ c 1 , c 2 ] , . . . , [ c n - 1 , c n ] , [ c n , b ]. In this case we simply define Z γ f = Z c 1 a f ( γ ( t )) γ 0 ( t ) dt + Z c 2 c 1 f ( γ ( t )) γ 0 ( t ) dt + · · · + Z b c n f ( γ ( t )) γ 0 ( t ) dt . In what follows, we’ll usually state our results for smooth curves, bearing in mind that practically all can be extended to piecewise smooth curves. Example 4.2. As our first example of the application of this definition we will compute the integral of the function f ( z ) = z 2 = ( x 2 - y 2 ) - i (2 xy ) over several curves from the point z = 0 to the point z = 1 + i . 42

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CHAPTER 4. INTEGRATION 43 (a) Let γ be the line segment from z = 0 to z = 1 + i . A parametrization of this curve is γ ( t ) = t + it, 0 t 1. We have γ 0 ( t ) = 1 + i and f ( γ ( t )) = ( t - it ) 2 , and hence Z γ f = Z 1 0 ( t - it ) 2 (1 + i ) dt = (1 + i ) Z 1 0 t 2 - 2 it 2 - t 2 dt = - 2 i (1 + i ) / 3 = 2 3 (1 - i ) . (b) Let γ be the arc of the parabola y = x 2 from z = 0 to z = 1 + i . A parametrization of this curve is γ ( t ) = t + it 2 , 0 t 1. Now we have γ 0 ( t ) = 1 + 2 it and f ( γ ( t )) = t 2 - ( t 2 ) 2 - i 2 t · t 2 = t 2 - t 4 - 2 it 3 , whence Z γ f = Z 1 0 ( t 2 - t 4 - 2 it 3 ) (1 + 2 it ) dt = Z 1 0 t 2 + 3 t 4 - 2 it 5 dt = 1 3 + 3 1 5 - 2 i 1 6 = 14 15 - i 3 . (c) Let γ be the union of the two line segments γ 1 from z = 0 to z = 1 and γ 2 from z = 1 to z = 1 + i . Parameterizations are γ 1 ( t ) = t, 0 t 1 and γ 2 ( t ) = 1 + it, 0 t 1. Hence Z γ f = Z γ 1 f + Z γ 2 f = Z 1 0 t 2 · 1 dt + Z 1 0 (1 - it ) 2 i dt = 1 3 + i Z 1 0 1 - 2 it - t 2 dt = 1 3 + i 1 - 2 i 1 2 - 1 3 = 4 3 + 2 3 i . The complex integral has some standard properties, most of which follow from their real siblings in a straightforward way. To state some of its properties, we first define the useful concept of the length of a curve. Definition 4.3. The length of a smooth curve γ is length( γ ) := Z b a γ 0 ( t ) dt for any parametrization γ ( t ), a t b , of γ .
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Chapter4 - Chapter 4 Integration Everybody knows that...

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