Doubt

[Goutham]

1.Sir,if f(z) and f(z bar) are
both analytic on a domain D,
then what can we say about.
f(z)
2.Let S be a set of vectors and
if L(S)=S,then what can the
set S be and Can we say that
Let V be a vector space then
for All such V, L(V)=V

[n17...@rguktn.ac.in]

1.If a function f and its conjugate are analytic on a domain D then we can say that f(z) is a constant function.

[RaJesH BanDari YaDav]

On Tuesday, September 22, 2020 at 8:41:55 AM UTC+5:30, Goutham wrote:

1.Sir,if f(z) and f(z bar) are
  both analytic on a domain D,
  then what can we say about.  
  f(z)

Ans:  Suppose f(z)=u+iv is an analytic function -----(1)

          given  conjugate of f is also analytic

          i.e u-iv is the analytic function ------(2)

           from (1) and (2) using C-R equations 

           u=constant and v=constant 

           f(z) is constant 

 

2.Let S be a set of vectors and
  if L(S)=S,then what can the  
  set S be and Can we say that
  Let V be a vector space then
  for All such V, L(V)=V

  

Ans:  Suppose let S be set of vectors ( either empty or non-empty set)

         We know that L(S) the liner Span of S is subspace (Vector Space) of  V

        

         and also L(L(S))=L(S)

         In the above question given  if L(S)=S  then S must be subspace of V or S=V

         L(V)=V is (TRUE)

 

[RaJesH BanDari YaDav]

On Tuesday, September 22, 2020 at 8:41:55 AM UTC+5:30, Goutham wrote:

1.Sir,if f(z) and f(z bar) are
  both analytic on a domain D,
  then what can we say about.  
  f(z)

    

<lakshmi tandyala>

Sir,for the first question it's not f bar ,what if f(z bar) which is different from (f(z)) bar

 <RaJeSH BanDari YaDav >  

 f(z)=f(x+iy)=u(x,y)+iv(x,y)

   f(bar(z))=f(x-iy)=f(x+i(-y))=u(x,-y)+iv(x,-y)

  

   given both analytic functions 

    from C-R equations 

    

    ux=vy   and uy=-vx   (' .' f(z) is analytic)-------(1)

    ux=-vy    and  uy=vx   (' .'  f(bar(z)) is analytic ) ) -----(2)

   from (1) and (2)   f= constant