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**ReviewSequence & Series of complex numbers First consider the useful facts**

The △ in-equality : If z_{1} & z_{2} are arbitrary complex no., then

**Proof: **

Taking positive sq. root yields the desired inequality.

Note:

Note equality occur in (i)

Note

Defn A seq. {zn} of complex no. is said to converge to a complex no. z if the seq. {|z_{n}- z|} of real no. converge to 0

**Proof forward Part :**

Assume that z_{n} → z then |zn- z|→ 0

Converse Part : Assume that Re(z_{n}) → Re(z) & Im(z_{n}) → Im(z)

consider,

|z_{n} - z|= |(x_{n +} iy_{n} )- (x+ iy)| where, z_{n} = x_{n} + iy_{n} & z = x + iy

Note: {z_{n}} can’t converge to more than one limit.

If exists is unique.

→ 0 as n→∞ (∵0 ≤|z|≤ 1)

**Def**^{n} A seq. {zn} of complex no. is called a Cauchy seq. if for each 𝜖 > 0, there exists (N=N(𝜖)) an integer N s.t.

Cauchy Criteria for convergence in complex plane {z_{n}} converges iff {z_{n}} is a Cauchy seq.

**Forward part**: Assume that z_{n} → z then

Re(zn) → Re(z) & Im(z_{n}) → Im(z)

Proved before

Here {Re(z_{n})} & {Im(z_{n})} both are Cauchy seq. being convergent seq. of real no.

Converse part: suppose {z_{m}} is a Cauchy seq. , then using

implies both {Re(z_{n}) & Im(z_{n})} are cauchy seq. of real no. hence both converges,

implies {z_{n}} converges

Def^{n} The infinite series of complex nos. is said to be converge if the seq. {S_{n}},

of paritial sum S is convergent.

from the Cauchy criterion, i.e. “a seq. is conv. iff it is a Cauchy seq”, we see that converges iff {S_{n}}is a cauchy seq.

For each

from this it follows that convergence of (i. e. a neccessary cond for the series to converge is that z_{n} →0 as n→ ∞

**Remark's** in the case of sequence we have, converges to z iff

converges to Re(z) converge to Im(z).

A sufficient cond^{n} for cgces of

i.e. absolutely convergent series is cgt.

also, as with real series, we say a series is said to be __absolutely convergent__ if the series +ve real no. is convergent. Futher using the fact that

Implies that “every absolutely convergent series is convergent.” Converse is not true e.g.

absolutely and hence cgt.

**Generalized Cauchy’s n ^{th} root test: **let

(a) if ℓ < 1 then series cgs abs.

(b) if ℓ >1 then series div.

(c) if ℓ = 1 the series may or may not conv.

**Generalised D’ lembert Ratio test:**** **then

(a) if L < 1, then the series cgs absolutely

(b) if l > 1, the series div.

(c)if l≤1≤ L, no conclusion.

**complex analysis ( Bak & Newmann)**

**Topology of the complex plane****1.5 Definition**

(i)** **is called an __open disc__ of radius r centered at z_{0} , also called nbhd of z_{0}.

(ii) circle

= circle with center z_{0} , radius r.

(iii) subset is called If for every , there exists r > 0 . It means that some disc arround z lies entirely in S. for instance, the interior of a circle the entire complex plane half plane (Re(z) , Im(z) <0 , Re(z) ) ect. are open sets. An __open disc__ is an open set.

(iv) A set is s. t. b open iff for each

Note: is __not__ open

Note: is open in but __not__ in

(v) set S is called closed if its complement

= Coll of points whose neighbourhood have a non empty intersection with both S and

(ix) = closure of S

(x) S is bdry iff for some r > 0.

(xi) S is compact iff S is closed bdd

(xii) set S is said to be __disconneced__ if there exists two disjoint open set s. t.

(xiii) S is s. t. b. disconnected iff S is a union of two non empty disjoit open subsets.

(xiv) S is called __connected__ if it is __not__ disconnected. in other words, S is connected if and only if, each pair of points z_{1}, z_{1} of S can be connected by __an arc__ lying in S

is calle the __line segment __with and points z_{1} & z_{2} and denoted by [z_{1},z_{2}]

∴ if for each then the line segment [z_{1}, z_{2}] , (where z_{1} , z_{2} ∈S ) is said to be containe in S.

→ by a __polygonal line__ from z_{1} to z_{n} … . a finite union of line segments of the form

(z_{1} and z_{n} are said to be __polygonally connected)__

→ A set S is said to be __polygonally connected__ if any two points of S can be connected by a __polygonal line (basically horizonal or verticaly) contained in S.__

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