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Complex Analysis

Analytic Continuation and Monodromy

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f(z) = z^{1/2}, around the unit circle starting at $z=1$

The function returns to its original value after one complete traversal, no monodromy since the path does not encircle the branch point at z = 0.

f(z) = (z-1)^{2/3}, around a path that starts at $z=2$ and encircles z = 1 once

Monodromy occurs, analytic continuation results in $e^{2\pi i/3}(z-1)^{2/3}$ after one complete traversal.

f(z) = \text{arg}(z), around a path that starts at $z=1$ and makes one full counter-clockwise loop around the origin.

The analytic continuation results in $\text{arg}(z) + 2\pi$ , reflecting monodromy.

f(z) = \sqrt[3]{z}, around a path that circles twice around the origin

After two complete traverses, monodromy occurs and the analytic continuation results in the same value of $\sqrt[3]{z}$ , as the argument increases by $4\pi$ .

f(z) = \arccos(z), around a path from $z=1$ along the real axis to $z=-1$ but just below the real axis, and back above the real axis to $z=1$

The function returns to its original value, no monodromy because the path does not cross the branch cut of arccos on the first traversal.

f(z) = \ln(z), around a circle of radius 2

Monodromy occurs, analytic continuation results in $f(z) + 2\pi i$ after one complete traversal.

f(z) = e^z, around the unit circle

The function returns to its original value after one complete traversal, no monodromy due to the periodic property of the exponential function.

f(z) = e^{\sqrt{z}}, starting at $z=1$ and around a path that encloses $z=-1$ but not the origin

The function is single-valued and it returns to its original value after one complete traversal; no monodromy occurs.

f(z) = \ln(z), around the unit circle

The function returns to its original value after one complete traversal, no monodromy since the path does not encircle the branch point at z = 0.

f(z) = \arcsin(z), starting from $z=0$ and traveling along a path just above the real axis to $z=-1$ and back to $z=0$

The function returns to its original value after the traversal; no monodromy because the path does not cross the branch cut of arcsin.

f(z) = \log(i z), around a path from $z=1$ to $z=i$ to $z=-1$ to $z=-i$ and back to $z=1$

Monodromy occurs, analytic continuation results in $\log(i z) + 2\pi i$ after one complete traversal.

f(z) = z^{3/2}, around a path from the positive real axis and looping around the origin once

Monodromy occurs, analytic continuation results in $-z^{3/2}$ after one complete traversal.

f(z) = z^{1/3}, around a circle of radius 1/2 centered at z = 1

The function returns to its original value after one complete traversal, no monodromy since the branch point at z = 0 is not encircled.

f(z) = z^{1/4}, around a path enclosing the origin

Monodromy occurs, analytic continuation results in $f(z)e^{\pi i/2}$ after one complete traversal, yielding a fourth root of unity different from the starting value.

f(z) = \log_{10}(z), around a loop from 1 to -1 below the real axis and back to 1 above the real axis

No monodromy occurs if the branch cut for the logarithm does not lie on the negative real axis; the function returns to its original value.

f(z) = \sqrt{z^2 - 1}, around a path encircling both branch points at z = 1 and z = -1

Monodromy occurs, analytic continuation results in $-\sqrt{z^2 - 1}$ after one complete traversal.

f(z) = \tan^{-1}(z), traveling from $z=0$ to $z=i$ and back to $z=0$ along paths parallel to the imaginary axis

The function returns to its original value after the traversal; no monodromy because the path does not cross the branch cuts of tan inverse.

f(z) = \text{Log}(z), where Log denotes the principal branch of the logarithm, around the unit circle

The function exhibits monodromy and the result of analytic continuation after one traversal is $\text{Log}(z) + 2\pi i$ .

f(z) = 1/z, around a path that loops around z = 0

The reciprocal function is single-valued; after encircling the origin, the function returns to its original value, no monodromy.

f(z) = \operatorname{erf}(z), where erf is the error function, around the unit circle

The function returns to its original value after one complete traversal since it is entire and thus no monodromy occurs.

f(z) = (z + i)^{1/2}, around a path encircling the branch point at z = -i

Monodromy occurs, analytic continuation results in $-(z + i)^{1/2}$ after one complete traversal.

f(z) = \sqrt{1-z^2}, looping around the point z = 0

The function returns to its original value after the traversal; no monodromy occurs because the path does not encircle the branch points at z = 1 and z = -1.

f(z) = \sinh^{-1}(z), traveling from $z=0$ along a path in the upper half-plane to $z=i$ and then back to $z=0$

The function returns to its original value, no monodromy because the path does not cross the branch cut of sinh inverse.

f(z) = e^{1/z}, around a small circle encircling the essential singularity at z = 0

Function values vary widely near an essential singularity and do not simply return to the original value; monodromy does not apply to essential singularities.

f(z) = \sqrt{z(z-1)(z-2)}, around a loop encircling z = 1 but not z = 0 or z = 2

Monodromy occurs, analytic continuation results in $-\sqrt{z(z-1)(z-2)}$ after one complete traversal around z = 1.

f(z) = \text{Log}(z+i), looping around the branch cut starting at z = -i

Monodromy occurs, analytic continuation leads to $\text{Log}(z+i) + 2\pi i$ after one complete traversal around the branch cut.

f(z) = \tan(z), around a loop from $z=0$ encircling the point $z=\frac{\pi}{2}$ but not $z=-\frac{\pi}{2}$

The function returns to its original value, no monodromy occurs because the path does not enclose a branch cut or extend to infinity.

f(z) = \frac{1}{z - i}, looping once around z = i

The function is single-valued, it returns to its original value, no monodromy since the loop encircles a pole, not a branch point.

f(z) = \log\left(\frac{z-1}{z+1}\right), looping from z = 2 around the branch cut joining the points at z = 1 and z = -1

Monodromy occurs, analytic continuation results in $\log\left(\frac{z-1}{z+1}\right) + 2\pi i$ after one complete traversal around the branch cut.

f(z) = \sqrt[4]{z-1}, around a path starting at $z=2$ that encircles z = 1 four times

After four complete traversals, the function returns to its original value; no monodromy because the argument cycles through $4\times\frac{\pi}{2}$ .

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