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Where is ##(z+1)Ln(z)## differentiable?

  • #1
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Homework Statement


Find the domain in which the complex-variable function ##f(z)=(z+1)Ln(z)## is differentiable. Note: ##Ln(z)## is the principal complex logarithmic function.

Homework Equations


Cuachy-Riemann Equations?

The Attempt at a Solution


The solution I have in mind would be to let ##z=x+iy## then substitute and simplify. Check if it satisfies the Cauchy-Riemann equations, the real and imaginary part of ##f## is continuous and their first-order partial derivative are continuous as well. But, I do not know how to simplify ##Arg(z)## in ##Ln(z)=Log_e(z)+iArg(z)## because ##z## is not fixed.
 

Answers and Replies

  • #2
MathematicalPhysicist
Gold Member
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It should be ##\log |z|## and not ##\log z##, i.e. natural logarithm of the modulus of z.
 
  • #3
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It should be ##\log |z|## and not ##\log z##, i.e. natural logarithm of the modulus of z.
Yes, I made a typo, but how do I simplify ##Arg(z)##?
 
  • #4
MathematicalPhysicist
Gold Member
4,220
172
Well, ##Arg(z)=\arctan y/x## where ##z=x+iy##, this should help you with Cauchy-Riemann.
 
  • #5
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  • #6
317
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It's what I have on paper but I can't reconcile with what wikipedia have. It has conditions depending on the values of x and y.
After giving it some thought now, it doesn't seem to matter when I start differentiating.
 

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