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Complex Analysis

Determime whether the following statement is true or false. Justify your answer.

If f is analytic in the unit disk Δ(0;1) and |f'(z)-1-i| < √2 for all z belongs to Δ(0;1). Then f is univalent in Δ(0;1).

If f is analytic in the unit disk Δ(0;1) and |f'(z)-1-i| < √2 for all z belongs to Δ(0;1). Then f is univalent in Δ(0;1).

Complex Analysis

Determine whether the statement is true or false. Justify the answer.

If f is analytic in a convex domain D such that Re f'(z) is not equal to 0 for all z belongs to D, then f is univalent in D

If f is analytic in a convex domain D such that Re f'(z) is not equal to 0 for all z belongs to D, then f is univalent in D

Complex Analysis

There exists an analytic univalent function f that maps the infinite strip {z : 0 < Im z < 1} onto the unit disk.

Complex Analysis

Determine whether the statement is true or false and Justify the answer.

If f is univalent and analytic in an open set D except for isolated singularities, then f can have at most one singularity and that as a simple pole.

If f is univalent and analytic in an open set D except for isolated singularities, then f can have at most one singularity and that as a simple pole.

Complex Analysis

Let f(z) = sin z/z

and f(0) = 0. Explain why f is analytic at z = 0. Find the Maclaurian

series for f(z) and g(z) = ∫ f(ξ)dξ from 0 to z

. Does there exist a function f with an

isolated singularity at 0 and such that |f(z)|~ exp( 1/|z|) near z= 0?

Complex Analysis

(a) Show that, for any complex number z, zz = |z|

2

, z + z = 2Re(z) and Re(z) ≤ |z|. Hence

show that

i. |z1 + z2|

2 = |z1|

2 + |z2|

2 + 2Re(z1z2),

ii. |z1 + z2| ≤ |z1| + |z2|,

where Re(z) is the real part of z and z the conjugate of z.

2

, z + z = 2Re(z) and Re(z) ≤ |z|. Hence

show that

i. |z1 + z2|

2 = |z1|

2 + |z2|

2 + 2Re(z1z2),

ii. |z1 + z2| ≤ |z1| + |z2|,

where Re(z) is the real part of z and z the conjugate of z.

Complex Analysis

Show that, for any complex number z, zz = |z|

2

, z + z = 2Re(z) and Re(z) ≤ |z|. Hence

show that

i. |z1 + z2|

2 = |z1|

2 + |z2|

2 + 2Re(z1z2),

ii. |z1 + z2| ≤ |z1| + |z2|,

where Re(z) is the real part of z and z the conjugate of z. [26 marks]

(b) If z1 = 1 + 2i, find the set of values of z2 for which

(i) |z1 + z2| = |z1| + |z2| (ii) |z1 + z2| = |z1| − |z2|.

2

, z + z = 2Re(z) and Re(z) ≤ |z|. Hence

show that

i. |z1 + z2|

2 = |z1|

2 + |z2|

2 + 2Re(z1z2),

ii. |z1 + z2| ≤ |z1| + |z2|,

where Re(z) is the real part of z and z the conjugate of z. [26 marks]

(b) If z1 = 1 + 2i, find the set of values of z2 for which

(i) |z1 + z2| = |z1| + |z2| (ii) |z1 + z2| = |z1| − |z2|.

Complex Analysis

z^3=6 ( cos ( π/3 ) + i sin ( π/63 ) )

Complex Analysis

Obtain the 6th root of (-7)

Complex Analysis

Suppose that f(z) is analytic/holomorphic in Ω, an open connected set and |f(z)| < 1 for z ∈ Ω. Show that the function defined by g(z) = Summation from n=1 to infinity n{f(z)}^n is also holomorphic in Ω