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


Analyticity of complex $operatornameLog(1-z^-4)$complex analysis - differentiabiliityproperties of analytic function $sin z$Finding the domain of analyticity of a logarithmConformal points, branch points, and analyticity of the mapping $z=w+e^w$?Help understanding branch cuts in inverse trig functions.Domain of analyticity for $log(f(z))$Domain of analyticity of $,sqrtz^2-a^2$Branch Cuts and Analyticity for $sqrt G(z)$, $z in mathbbC$What is the domain of analyticity for $G(w) = sinsqrt w$?













4












$begingroup$


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 !










share|cite|improve this question











$endgroup$







  • 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















4












$begingroup$


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 !










share|cite|improve this question











$endgroup$







  • 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













4












4








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 !










share|cite|improve this question











$endgroup$




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















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 26 at 8:46







topspin

















asked Mar 21 at 6:49









topspintopspin

660413




660413







  • 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












  • 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







1




1




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




$begingroup$
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




$begingroup$
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




$begingroup$
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




$begingroup$
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




$begingroup$
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










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