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Question Number 2874 by RasheedAhmad last updated on 29/Nov/15
What is the meaning of  (i) z→0   (ii) z→z_0    z,z_0 ∈C
Whatisthemeaningof(i)z0(ii)zz0z,z0C
Answered by Filup last updated on 29/Nov/15
(1)  z approaches 0           z→0^+  approaches from right side           z→0^−  approaches from left side  e.g.  f(x)= { ((x≥0,  1)),((x<0,  2)) :}  lim_(x→0^+ ) f(x)=1  lim_(x→0^− ) f(x)=2    (2)  z, z_0 ∈C  means that both z and z_0   are within the complex set. Simply,  they are complex functions.  Such as:   z=1+2i    if  z→z_0 , then for complex z, it   approaches complex z_0
(1)zapproaches0z0+approachesfromrightsidez0approachesfromleftsidee.g.f(x)={x0,1x<0,2limx0+f(x)=1limx0f(x)=2(2)z,z0Cmeansthatbothzandz0arewithinthecomplexset.Simply,theyarecomplexfunctions.Suchas:z=1+2iifzz0,thenforcomplexz,itapproachescomplexz0
Answered by Yozzi last updated on 29/Nov/15
For z∈C  defined simply as z=x+iy  where x,y∈R ,  (i)z→0⇒x+iy→0⇒(x→0)∧(y→0)  (ii)z→z_0 ⇒x+iy→x_0 +iy_0   ⇒(x→x_0 )∧(y→y_0 ).    E.g Define f(z)=z^2 , z∈C.  ⇒lim_(z→0) f(z)=lim_(z→0) z^2 =0^2 =0  But, if z=x+iy, x,y∈R,  f(x)=(x+iy)^2 =x^2 −y^2 +2iyx  ∴lim_(z→0) f(z)=lim_((x,y)→(0,0)) (x^2 −y^2 +2ixy)                     =0^2 −0^2 +2i×0×0                     =0    Similarly, suppose z_0 =3+i.  lim_(z→z_0 ) f(z)=lim_(z→z_0 ) z^2 =(3+i)^2 =8+6i  Alternatively,  lim_(z→z_0 ) f(z)=lim_((x,y)→(3,1)) x^2 −y^2 +2ixy                 =9−1+2i×1×3                 =8+6i
ForzCdefinedsimplyasz=x+iywherex,yR,(i)z0x+iy0(x0)(y0)(ii)zz0x+iyx0+iy0(xx0)(yy0).E.gDefinef(z)=z2,zC.limz0f(z)=limz0z2=02=0But,ifz=x+iy,x,yR,f(x)=(x+iy)2=x2y2+2iyxlimz0f(z)=lim(x,y)(0,0)(x2y2+2ixy)=0202+2i×0×0=0Similarly,supposez0=3+i.limzz0f(z)=limzz0z2=(3+i)2=8+6iAlternatively,limzz0f(z)=lim(x,y)(3,1)x2y2+2ixy=91+2i×1×3=8+6i
Commented by Yozzi last updated on 29/Nov/15
I see. Thanks!
Isee.Thanks!
Commented by 123456 last updated on 29/Nov/15
however there infinite ways to aproach  z_0  on C, so if the limit depends on path  that you taked, them it doenst exist  [like in the R, lim_(x→0^− ) f(x)≠lim_(x→0^+ ) f(x),lim_(x→0) f(x)=∄]
howeverthereinfinitewaystoaproachz0onC,soifthelimitdependsonpaththatyoutaked,themitdoenstexist[likeintheR,limx0f(x)limx0+f(x),limx0f(x)=]
Commented by prakash jain last updated on 29/Nov/15
The same concept (path dependency applies)  when computing limits for functions of  more than one variable.  for z→0 or z→z_0  limits it is sometimes easier to  convert to polar coordinates then consider  limit on r with no constraints on θ.
Thesameconcept(pathdependencyapplies)whencomputinglimitsforfunctionsofmorethanonevariable.forz0orzz0limitsitissometimeseasiertoconverttopolarcoordinatesthenconsiderlimitonrwithnoconstraintsonθ.
Commented by Rasheed Soomro last updated on 29/Nov/15
What are other ways, for example, of approaching z_0   and what is ′path dependancy′?
Whatareotherways,forexample,ofapproachingz0andwhatispathdependancy?
Commented by 123456 last updated on 29/Nov/15
lets z_0 =0  we can aproach z by  z=te^(ıt) ,t→0^+  (path 1)    z=re^(ıθ) ,r→0 (path 2)  the first path is a spiral and the second  is a stray line with angle θ with real line  path depedency=limits depends on the  path you take to aproach z_0   in real case with one varriable we have  only the paths  x→0^+   x→0^−   however in C we have infinites path  as you see in above a example of two  path, there are more paths that you  can take.
letsz0=0wecanaproachzbyz=teıt,t0+(path1)z=reıθ,r0(path2)thefirstpathisaspiralandthesecondisastraylinewithangleθwithreallinepathdepedency=limitsdependsonthepathyoutaketoaproachz0inrealcasewithonevarriablewehaveonlythepathsx0+x0howeverinCwehaveinfinitespathasyouseeinaboveaexampleoftwopath,therearemorepathsthatyoucantake.
Commented by Rasheed Soomro last updated on 29/Nov/15
T HANK^(Sss)  for all of YOU !  I always have learnt from you.
THANKSssforallofYOU!Ialwayshavelearntfromyou.

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