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

In summary, the problem asks to find the domain in which the complex-variable function ##f(z)=(z+1)Ln(z)## is differentiable, where ##Ln(z)## is the principal complex logarithmic function. The suggested solution is to substitute ##z=x+iy##, simplify, and check if it satisfies the Cauchy-Riemann equations. However, there is uncertainty regarding how to simplify ##Arg(z)##, with suggestions including using ##\arctan y/x## or referencing Wikipedia for conditions depending on the values of x and y. After further consideration, it is concluded that the differentiation process does not depend on the starting point.
  • #1
Terrell
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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.
 
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  • #2
It should be ##\log |z|## and not ##\log z##, i.e. natural logarithm of the modulus of z.
 
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  • #3
MathematicalPhysicist said:
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
Well, ##Arg(z)=\arctan y/x## where ##z=x+iy##, this should help you with Cauchy-Riemann.
 
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  • #5
MathematicalPhysicist said:
Well, ##Arg(z)=\arctan y/x## where ##z=x+iy##, this should help you with Cauchy-Riemann.
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. https://en.wikipedia.org/wiki/Argument_(complex_analysis)
 
  • #6
Terrell said:
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.
 

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

1. What is the definition of differentiability?

Differentiability is a mathematical concept that describes the smoothness or continuity of a function. A function is said to be differentiable at a point if the slope of the tangent line to the graph of the function at that point exists and is finite.

2. Is ##(z+1)Ln(z)## continuous?

Yes, ##(z+1)Ln(z)## is continuous. This means that the function has no sudden jumps or breaks in its graph and can be drawn without lifting your pencil.

3. What is the derivative of ##(z+1)Ln(z)##?

The derivative of ##(z+1)Ln(z)## is ##Ln(z)+\frac{z+1}{z}##. This can be found using the product rule and the chain rule.

4. Is ##(z+1)Ln(z)## differentiable for all values of z?

No, ##(z+1)Ln(z)## is not differentiable for all values of z. It is not differentiable at z = 0 because the derivative at this point is undefined. It is also not differentiable at any negative values of z.

5. How can I determine if ##(z+1)Ln(z)## is differentiable at a specific point?

To determine if ##(z+1)Ln(z)## is differentiable at a specific point, you can find the derivative of the function and evaluate it at that point. If the derivative exists and is finite, then the function is differentiable at that point.

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