Cut point in LOG smoothly

Aug 6th, 2022
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How to Cut point in LOG

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greeting students and welcome back to another lesson on complex variables in this video Im going to define branch points and branch cuts and build a foundation thatll allow us to perform complex integration of functions with branches to build this foundation Ill start by discussing the natural log of a complex number Z suppose I have a function W equals F of Z which equals lon of Z if you remember my very first video on complex variables youll recall that I can write any complex number as its real part plus I times its imaginary part in the case of W I can write it as u plus IV where u and v are real numbers I can also write a complex number Z in the complex plane as a polar representation with Z being the distance from the origin R times the exponential of I times the angle relative to the positive real axis theta in that case the natural log of Z will become the natural log of R plus I times theta note that R and theta are both positive real numbers if I now rearrange this equat

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A branch point is a point from which the signal from a block goes concurrently to other blocks or summing points.
For example, the function w = z1/2 has two branches: one where the square root comes in with a plus sign, and the other with a minus sign. A branch cut is a curve in the complex plane such that it is possible to define a single analytic branch of a multi-valued function on the plane minus that curve.
A branch point of an analytic function is a point in the complex plane whose complex argument can be mapped from a single point in the domain to multiple points in the range.
The values of z that make the expression under the square root zero will be branch points; that is, z=±i are branch points. Let z−i=r1eiθ1 and z+i=r2eiθ2. Then f(z)=√z2+1=√r1r2ei(θ1+θ2)/2.
A branch cut is a curve (with ends possibly open, closed, or half-open) in the complex plane across which an analytic multivalued function is discontinuous. For convenience, branch cuts are often taken as lines or line segments. Branch cuts (even those consisting of curves) are also known as cut lines (Arfken 1985, p.
A branch point of an analytic function is a point in the complex plane whose complex argument can be mapped from a single point in the domain to multiple points in the range.
cutpoint (plural cutpoints) (topology) A point of a connected space whose removal causes the resulting space to be disconnected. Every point of a line is a cutpoint, but no point of a circle is a cutpoint.
7:23 9:30 But I didn't need to have a branch cut here specifically. I could have it here or here the point isMoreBut I didn't need to have a branch cut here specifically. I could have it here or here the point is that all of these branch cuts share the origin in common. And this common point is called the branch
In the case of an analytic function of several complex variables f(z), z=(z1… zn), n≥2, a point a of the space Cn or CPn is said to be a branch point of order m, 1≤m≤∞, if it is a branch point of order m of the, generally many-sheeted, domain of holomorphy of f(z).
The branch cut for fα is the ray r ≥ 0, θ = α, which includes the origin. The point z = 0, common to all branch cuts for the multivalued square root function, is called a branch point. The mapping w = fα (z) and its branch cut are illustrated in Figure 2.19.

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