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# The Line Integral Essay

575 words - 3 pages

EXERCISE-04

The line integral

Evaluate [pic]

a) If [pic] and C

i) is the line segment from z = 0 to z = 1+i

ii) consists of two line segments, one from z = 0 to z = i and other from z = i to z = i+1.

b) If f(z) = z2 and C is the line segment from z = 0 to z = 2+ i

c) If f(z) = z2 and C consists of two line segments, one from z = 0 to z = 2 and other from z = 2 to z = 2+i.

d) If f(z) = 3z + 1 and C follows the figure

e) If [pic] , C is a circle [pic]and [pic]

f) If [pic] and the path of integration C is the upper half of the circle [pic] from z = -1 to z = 1.

g) If [pic] and C is

Evaluate [pic] where C is the circle
[pic].

8. Evaluate [pic] where C is
(a) [pic]

(b) the square with vertices at [pic]

9. Evaluate [pic] around the square with vertices at
[pic]

10. Evaluate [pic] where C is the square with vertices at
[pic]

11. Evaluate by Cauchy’s integral formulae and by Residue theorem

[pic]
[pic]
[pic]
[pic]

EXERCISE-06

1. State Taylor’s and Laurent’s theorems.
2. Show that [pic]
3. Show that [pic]; [pic]
4. Show that [pic]; [pic]

5. Obtain Laurent series expansion of
[pic] when [pic], [pic]
[pic] when [pic],

6. Expand [pic]in a Laurent series valid for

[pic].

7. Expand [pic]in a Laurent series valid for

[pic]

8. Expand [pic]in a Laurent series valid for

[pic]

9. If [pic], find a Laurent series of f(z) about [pic] convergent
for [pic]
10. Find the zeros...

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