Question

in problem given the points at which the given function will not be analytic f(z)=z/z-3i​

Answer 1

Answer:

We have

1 − i

2

=

1

2

+ (−

1

2

)i.

So a =

1

2

and b = −

1

2

.

5. We have

(2 − i)

2 = (2 + i)

2

(because 2 − i = 2 − (−i) = 2 + i)

= 4 + 4i +

=−1

z}|{

(i)

2 = 3 + 4i.

So a = 3 and b = 4.

9. We have

1

2

+

i

7

3

2

− i

=

1

2

3

2

+ i

3

14

−

1

2

= 1

7

z}|{

−i

i

7

=

25

28

− i

2

7

So a =

25

28 and b = −

2

7

.

13. Multiplying and dividing by the conjugate of the denominator, i.e. by 2 − i = 2 + i we

get

14 + 13i

2 − i

=

(14 + 13i)(2 + i)

(2 − i)(2 + i)

=

14 · 2 + 14 · i + 13i · 2 + 13i

2

4 + 1

=

28 + 14i + 26i − 13

5

=

15

5

+

40

5

i = 3 + 8i

So a = 3 and b = 8.

17. Multiplying and dividing by the conjugate of the denominator, i.e. by x − iy = x + iy

we get

x + iy

x − iy

=

x + iy

x − iy

·

(x + iy)

(x + iy)

=

(x + iy)

2

x

2 + y

2

=

x

2 + 2xyi + y

2

i

2

x

2 + y

2

=

x

2 − y

2 + 2xyi

x

2 + y

2

=

x

2 − y

2

x

2 + y

2

+

2xy

x

2 + y

2

i.