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 Defnition and Basic Properties At frst sight, complex integration is not really anything diFerent ±rom real integration. ²or a continuous complex-valued ±unction φ :[ a, b ] R C ,wedefne ° b a φ ( t ) dt = ° b a Re φ ( t ) dt + i ° b a Im φ ( t ) dt . (4.1) ²or a ±unction which takes complex numbers as arguments, we integrate over a curve γ (instead o± a real interval). Suppose this curve is parametrized by γ ( t ) ,a t b . I± one meditates about the substitution rule ±or real integrals, the ±ollowing defnition, which is based on (4.1) should come as no surprise. Defnition 4.1. Suppose γ is a smooth curve parametrized by γ ( t ) ,a t b , and f is a complex ±unction which is continuous on γ .Thenw edefneth e integral of f on γ as ° γ f = ° γ f ( z ) dz = ° b a f ( γ ( t )) γ ° ( t ) dt . This defnition can be naturally extended to piecewise smooth curves, that is, those curves γ whose parametrization γ ( t ), a t b , is only piecewise diFerentiable, say γ ( t ) is diFerentiable on the intervals [ a, c 1 ] , [ c 1 ,c 2 ] ,..., [ c n 1 ,c n ] , [ c n ,b ]. In this case we simply defne ° γ f = ° c 1 a f ( γ ( t )) γ ° ( t ) dt + ° c 2 c 1 f ( γ ( t )) γ ° ( t ) dt + ··· + ° b c n f ( γ ( t )) γ ° ( t ) dt . In what ±ollows, we’ll usually state our results ±or smooth curves, bearing in mind that practically all can be extended to piecewise smooth curves. Example 4.2. As our frst example o± the application o± this defnition we will compute the integral o± the ±unction f ( z )= z 2 = ± x 2 y 2 ² i (2 xy ) over several curves ±rom the point z = 0 to the point z =1+ i
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CHAPTER 4. INTEGRATION 43 (a) Let γ be the line segment from z =0t o z =1+ i . A parametrization of this curve is γ ( t )= t + it, 0 t 1. We have γ ° ( t )=1+ i and f ( γ ( t )) = ( t it ) 2 , and hence ° γ f = ° 1 0 ( t it ) 2 (1 + i ) dt =(1+ i ) ° 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 =0to z =1+ i . A parametrization of this curve is γ ( t )= t + it 2 , 0 t 1. Now we have γ ° ( t )=1+2 it and f ( γ ( t )) = ± t 2 ² t 2 ³ 2 ´ i 2 t · t 2 = t 2 t 4 2 it 3 , whence ° γ f = ° 1 0 ² t 2 t 4 2 it 3 ³ (1 + 2 it ) dt = ° 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 =0to z = 1 and γ 2 from z =1to z =1+ i . Parameterizations are γ 1 ( t )= t, 0 t 1 and γ 2 ( t )=1+ it, 0 t 1. Hence ° γ f = ° γ 1 f + ° γ 2 f = ° 1 0 t 2 · 1 dt + ° 1 0 (1 it ) 2 idt = 1 3 + i ° 1 0 1 2 it t 2 dt = 1 3 + i µ 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 Frst deFne the useful concept of the
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This note was uploaded on 02/27/2012 for the course MATH 417 taught by Professor Staff during the Fall '11 term at SUNY Albany.

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Chapter4 - Chapter 4 Integration Everybody knows that...

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