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QUESTION: 1

The integration of f (z) = x^{2} + ixy from A(1, 1) to B(2, 4) along the straight line AB joining the two points is

Solution:

x = at + b, y = ct + d

QUESTION: 2

Solution:

We know by the derivative of an analytic function that

QUESTION: 3

Solution:

QUESTION: 4

where c is the upper half of the circle z = 1

Solution:

Given contour c is the circle |z| = 1

QUESTION: 5

Solution:

Let f (z) = cosπz then f(z) is analytic within and on |z| =3, now by Cauchy’s integral formula

QUESTION: 6

Solution:

QUESTION: 7

The value of around a rectangle with vertices at is

Solution:

QUESTION: 8

where c is the circle x^{2} + y^{2} = 4

Que: The value of f(3) is

Solution:

QUESTION: 9

where c is the circle x^{2} + y^{2} = 4

Que: The value of f' (1 - i) is

Solution:

The point (1 - i) lies within circle |z| = 2 ( ... the distance of 1 - i i.e., (1, 1) from the origin is √2 which is less than 2, the radius of the circle).

Let Ø(z) = 3z^{2} + 7z + 1 then by Cauchy’s integral formula

QUESTION: 10

Expand the given function in Taylor’s series.

Que:

Solution:

QUESTION: 11

Expand the given function in Taylor’s series

Que:

Solution:

QUESTION: 12

Expand the given function in Taylor’s series.

Que:

Solution:

QUESTION: 13

If |z + 1| < 1, then z^{-2 }is equal to

Solution:

QUESTION: 14

Expand the function in Laurent’s series for the condition given in question.

Que: 1 < |z| < 2

Solution:

QUESTION: 15

Expand the function in Laurent’s series for the condition given in question.

Que: |z| > 2

Solution:

### Analytic functions - Complex variables

Doc | 2 Pages

### Part I: Complex Variables, Lec 5: Integrating Complex Functions

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### Part I: Complex Variables, Lec 1: The Complex Numbers

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### Part I: Complex Variables, Lec 2: Functions of a Complex Variable

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