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Is the complex logarithm an example of a function analytic on the domain $1<|z|<2$ such that it's derivative on the domain is $\frac{1}{z}$?
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Fermat said:Is the complex logarithm an example of a function analytic on the domain $1<|z|<2$ such that it's derivative on the domain is $\frac{1}{z}$?
Thanks
Fermat said:Is the complex logarithm an example of a function analytic on the domain $1<|z|<2$ such that it's derivative on the domain is $\frac{1}{z}$?
Thanks
Prove It said:No, the complex logarithm is never analytic on the negative real axis...
No. The complex logarithm can be defined as an analytic function locally in the neighbourhood of any nonzero complex number. But it cannot be defined globally as an analytic function on the annulus $1<|z|<2$. In fact, it cannot even be defined as a continuous function on that annulus.Fermat said:Is the complex logarithm an example of a function analytic on the domain $1<|z|<2$ such that it's derivative on the domain is $\frac{1}{z}$?
Opalg said:No. The complex logarithm can be defined as an analytic function locally in the neighbourhood of any nonzero complex number. But it cannot be defined globally as an analytic function on the annulus $1<|z|<2$. In fact, it cannot even be defined as a continuous function on that annulus.
Opalg said:No. The complex logarithm can be defined as an analytic function locally in the neighbourhood of any nonzero complex number. But it cannot be defined globally as an analytic function on the annulus $1<|z|<2$. In fact, it cannot even be defined as a continuous function on that annulus...
... somewhere along the route, there has to be a discontinuity. It could be on the negative real axis, as Prove It wants, or on the positive real axis, or at some other point. But there has to be a discontinuity somewhere, and consequently a point where $\log z$ fails to be analytic...
The basic difficulty here is that "$\log z$" is not really a function at all, in the strict sense of the word. A function by definition must be single-valued. But $\log z = \ln|z| + i\arg z$, and $\arg z$ is not single-valued: it is only defined modulo a multiple of $2\pi$. So if you want to define $\log z$ as an analytic function in some domain, you need to specify which branch of the $\arg$ function to use throughout that domain. In a simply-connected domain like the disk of radius $\sqrt2$ centred at $z=-1+i$, that can be done (in that case, by choosing the value of $\arg z$ that lies between $0$ and $2\pi$), but in a domain that includes a circuit of the origin, it is not possible to give a choice of $\arg z$ that is continuous throughout the domain.chisigma said:Very well!... it is clear that, since the function ln z analytic in every point of the real positive axis, on it the function is continuous and differentiable. If, therefore, there must be 'somewhere' discontinuity, we can take into account the positive imaginary axis, the negative real axis (as Prove It wants), and finally the negative imaginary axis... there seems no other reasonable choices. Now let's pick a point in the region indicated by Fermat where ln z is 'locally analytic', the point z = -1 + i. Let us look at the following figure...
https://www.physicsforums.com/attachments/2778In z= -1 + i ln z is analytic, i.e. it is expressed as Taylor series that converges inside a circle and everywhere inside this circle the function is analytic. Çlearly on the segment joining the point z = -1 + i and the point z = 0, the function is analytic, so that the circle of convergence is the one shown in the figure. But within the circle there are parts of the positive imaginary axis and parts of the negative real axis , so that both can not be the 'boundaries of discontinuities' we are looking for. If we develop ln z around the point z = -1-i we come to also exclude the negative imaginary axis, so that this 'border of discontinuity' seems to be a little like the the 'Araba Fenice' ... everyone says that it is but nobody knows where it is (Thinking)...
Kind regards
$\chi$ $\sigma$
Opalg said:The basic difficulty here is that "$\log z$" is not really a function at all, in the strict sense of the word. A function by definition must be single-valued. But $\log z = \ln|z| + i\arg z$, and $\arg z$ is not single-valued: it is only defined modulo a multiple of $2\pi$. So if you want to define $\log z$ as an analytic function in some domain, you need to specify which branch of the $\arg$ function to use throughout that domain. In a simply-connected domain like the disk of radius $\sqrt2$ centred at $z=-1+i$, that can be done (in that case, by choosing the value of $\arg z$ that lies between $0$ and $2\pi$), but in a domain that includes a circuit of the origin, it is not possible to give a choice of $\arg z$ that is continuous throughout the domain.
chisigma said:True to the saying of Dante Alighieri I have chosen as signature I decide not to waste any more time on this topic. Sooner or later I will open a math note that validates incontrovertibly the concept I said (Wave)...
Kind regards
$\chi$ $\sigma$
I like Serena said:I'll be looking forward to this incontrovertible validation!
(Sorry. I just felt an irresistible need to post.)
A complex logarithm is a mathematical function that is defined for complex numbers. It is an extension of the real logarithm function, which is defined only for positive real numbers. The complex logarithm is defined for all complex numbers except 0, and it is expressed as log(z) or ln(z).
An analytic function is a complex function that is differentiable at every point in its domain. This means that it has a well-defined derivative at every point, and can be represented by its Taylor series expansion. In other words, an analytic function is a smooth, continuously differentiable function that can be approximated by polynomials.
The domain of the complex logarithm function is the entire complex plane, except for the point z=0. This means that the function is defined for all complex numbers except for the origin. However, it is often only considered in the region 1<|z|<2, as it is not single-valued outside of this region.
The complex logarithm is considered an analytic function on the region 1<|z|<2 because it is differentiable at every point within this region. This means that it satisfies the Cauchy-Riemann equations, and can be expressed as a Taylor series. However, it is not analytic outside of this region, as it is not single-valued.
The complex logarithm function is used in various areas of mathematics, including complex analysis, number theory, and physics. It has applications in solving equations involving complex numbers, calculating complex integrals, and describing the behavior of physical systems. It is also used in the study of fractals and chaotic systems.