Page 45 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 4.5 Keywords

Calculus integrals: If  : D  C is simply a function on a real interval D = [, ], then the integral

rd
  (t)dt of course, simply an ordered pair of everyday 3 grade calculus integrals:

  
  (t)dt   x(t)dt i y(t)dt.


  
Antiderivative: Suppose f : D  C is continuous, where D is connected and every point of D is an
interior point. Then f has an antiderivative if and only if the integral between any two points of
D is path independent.

4.6 Self Assessment


1. If  : D  C is simply a function on a real interval D = [, ], then the integral  (t)dt of


  
rd



course, simply an ordered pair of everyday 3 grade .................  (t)dt   x(t)dt i y(t)dt.
  
2. A Riemann sum associated with the partition P is just what it is in the real case: .................
where z is a point on the arc between z and z, and zj = z – z .
*
j–1
j
j–1
j
j
3. Integral depends only on the points a and b and not at all on the path C. We say the integral
is .................
4. If in D the integrand f is the derivative of a function F, then any integral ................. for C 
D is path independent.
5. f is continuous at z, and so ................. Hence,

lim F(z   z) F(z)  f(z) = lim   1  (f(s) f(z))ds 

 z 0  z  z   z 
0
 L  z 
= 0
6. Suppose f : D  C is continuous, where D is connected and every point of D is an interior
point. Then f has an ................. if and only if the integral between any two points of D is
path independent.

4.7 Review Questions

1. Evaluate the integral zdz, where C is the parabola y = x from 0 to 1 + i.

2
C
1
2. Evaluate  dz, where C is the circle of radius 2 centered at 0 oriented counter clockwise.
C z


3. Evaluate f(z)dz, where C is the curve y = x from –1 – i to 1 + i , and f(z)   1 for y  0

3
C  4y for y  0

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