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Analyticity of this function $sqrtcoth^2(a z) + coth^2(b z) - c$

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4

1

$begingroup$

I want to determine the domain of analyticity of this function:
$$f(z) =sqrtcoth^2(a z) + coth^2(b z) - c$$
And $$c in ]0,1]$$

Where $$a,b in mathbbZ - mathbbZ^+$$ and $a , b$ finite say $a,b in [-1000 , 0]$

---

Suppose instead i took that $$f(z)=sqrtcoth^2(- z) + coth^2(-3 z) - 1$$

Letting $$w=coth^2(- z) + coth^2(-3 z) - 1$$

And finding when $w=0$ is that engough to find the branch points .

Is that enugh or we must see how is $Arg(coth^2(- z) + coth^2(-3 z) - 1)$ behave ?

If so , How can i see that ?

Ok , I Don't see where is the problem ?

If for example we have $$f(z)=sqrtcoth(z)=sqrtfraccosh(z)sinh(z)=sqrt1+frac2e^2z-1$$

We get Roots $$ z= frac12 i(2pi n + pi) , z in mathbbZ$$

Does that mean that we have infinite branch points ? including 0

So Where is the $sqrtcoth(z) $ is Analytic ?

Than you !

real-analysis complex-analysis analytic-functions

share|cite|improve this question

edited Mar 26 at 8:46

topspin

asked Mar 21 at 6:49

topspintopspin

660413

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Is Coth the hyperbolic cotangent? (maybe dumb question...)
  $endgroup$
  – coffeemath
  Mar 21 at 8:00
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes it is the hyperbolic cotangent .
  $endgroup$
  – topspin
  Mar 21 at 8:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Square roots of negatives are OK in the complex numbers, provided one takes care about branches...
  $endgroup$
  – coffeemath
  Mar 21 at 9:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  So , Is this function analytic in the first Quadrant ?
  $endgroup$
  – topspin
  Mar 21 at 9:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I don't know... what branch of square root chosen would make a difference I guess, but I'm no expert on that.
  $endgroup$
  – coffeemath
  Mar 21 at 9:39
  

  
  
  
  

 | 
show 2 more comments

4

1

$begingroup$

I want to determine the domain of analyticity of this function:
$$f(z) =sqrtcoth^2(a z) + coth^2(b z) - c$$
And $$c in ]0,1]$$

Where $$a,b in mathbbZ - mathbbZ^+$$ and $a , b$ finite say $a,b in [-1000 , 0]$

---

Suppose instead i took that $$f(z)=sqrtcoth^2(- z) + coth^2(-3 z) - 1$$

Letting $$w=coth^2(- z) + coth^2(-3 z) - 1$$

And finding when $w=0$ is that engough to find the branch points .

Is that enugh or we must see how is $Arg(coth^2(- z) + coth^2(-3 z) - 1)$ behave ?

If so , How can i see that ?

Ok , I Don't see where is the problem ?

If for example we have $$f(z)=sqrtcoth(z)=sqrtfraccosh(z)sinh(z)=sqrt1+frac2e^2z-1$$

We get Roots $$ z= frac12 i(2pi n + pi) , z in mathbbZ$$

Does that mean that we have infinite branch points ? including 0

So Where is the $sqrtcoth(z) $ is Analytic ?

Than you !

real-analysis complex-analysis analytic-functions

share|cite|improve this question

edited Mar 26 at 8:46

topspin

asked Mar 21 at 6:49

topspintopspin

660413

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Is Coth the hyperbolic cotangent? (maybe dumb question...)
  $endgroup$
  – coffeemath
  Mar 21 at 8:00
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes it is the hyperbolic cotangent .
  $endgroup$
  – topspin
  Mar 21 at 8:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Square roots of negatives are OK in the complex numbers, provided one takes care about branches...
  $endgroup$
  – coffeemath
  Mar 21 at 9:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  So , Is this function analytic in the first Quadrant ?
  $endgroup$
  – topspin
  Mar 21 at 9:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I don't know... what branch of square root chosen would make a difference I guess, but I'm no expert on that.
  $endgroup$
  – coffeemath
  Mar 21 at 9:39
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

I want to determine the domain of analyticity of this function:
$$f(z) =sqrtcoth^2(a z) + coth^2(b z) - c$$
And $$c in ]0,1]$$

Where $$a,b in mathbbZ - mathbbZ^+$$ and $a , b$ finite say $a,b in [-1000 , 0]$

---

Suppose instead i took that $$f(z)=sqrtcoth^2(- z) + coth^2(-3 z) - 1$$

Letting $$w=coth^2(- z) + coth^2(-3 z) - 1$$

And finding when $w=0$ is that engough to find the branch points .

Is that enugh or we must see how is $Arg(coth^2(- z) + coth^2(-3 z) - 1)$ behave ?

If so , How can i see that ?

Ok , I Don't see where is the problem ?

If for example we have $$f(z)=sqrtcoth(z)=sqrtfraccosh(z)sinh(z)=sqrt1+frac2e^2z-1$$

We get Roots $$ z= frac12 i(2pi n + pi) , z in mathbbZ$$

Does that mean that we have infinite branch points ? including 0

So Where is the $sqrtcoth(z) $ is Analytic ?

Than you !

real-analysis complex-analysis analytic-functions

share|cite|improve this question

edited Mar 26 at 8:46

topspin

asked Mar 21 at 6:49

topspintopspin

660413

$endgroup$

share|cite|improve this question

edited Mar 26 at 8:46

topspin

asked Mar 21 at 6:49

topspintopspin

660413

share|cite|improve this question

edited Mar 26 at 8:46

topspin

asked Mar 21 at 6:49

topspintopspin

660413

asked Mar 21 at 6:49

topspintopspin

660413

asked Mar 21 at 6:49

topspintopspin

660413