Courses

# Doc- Review: Sequence & Series of complex numbers Mathematics Notes | EduRev

## Mathematics : Doc- Review: Sequence & Series of complex numbers Mathematics Notes | EduRev

The document Doc- Review: Sequence & Series of complex numbers Mathematics Notes | EduRev is a part of the Mathematics Course Topic-wise Tests & Solved Examples for IIT JAM Mathematics.
All you need of Mathematics at this link: Mathematics

Review
Sequence & Series of complex numbers
First consider the useful facts

The △ in-equality : If z1 & z2 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. {|zn- z|} of real no. converge to 0

Proof forward Part :
Assume that zn → z then |zn- z|→ 0

Converse Part : Assume that Re(zn) → Re(z)  &  Im(zn) → Im(z)

consider,

|zn -  z|= |(xn + iyn )- (x+ iy)| where, zn = xn + iyn  & z = x + iy

Note: {zn} can’t converge to more than one limit.
If exists is unique.

→ 0 as n→∞ (∵0 ≤|z|≤ 1)

Defn 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 {zn} converges iff {zn} is a Cauchy seq.

Forward part: Assume that zn → z then

Re(zn) → Re(z)   & Im(zn) → Im(z)

Proved before
Here {Re(zn)} & {Im(zn)} both are Cauchy seq. being convergent seq. of real no.

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

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

implies {zn} converges
Defn The infinite series  of complex nos. is said to be converge if the seq. {Sn},

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 {Sn}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 zn →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 condn 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 nth root test: let   be a series of complex term such that

(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 z0 , also called nbhd of z0.

(ii) circle

= circle with center z0 , 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 z1, z1 of S can be connected by an arc lying in S

is calle the line segment with and points z1 & z2 and denoted by [z1,z2]

∴ if  for each  then the line segment [z1, z2] , (where z1 , z2 ∈S ) is said to be containe in S.
→ by a polygonal line from z1 to zn … . a finite union of line segments of the form

(z1 and zn 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.

Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

## Topic-wise Tests & Solved Examples for IIT JAM Mathematics

27 docs|150 tests

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

;