Today I Learned

Some of the things I've learned every day since Oct 10, 2016

Category Archives: analysis

183: Cauchy’s Argument Principle

In complex analysis, Cauchy’s Argument Principle relates the value of the contour integral of a meromorphic function along a closed curve to the number of poles and zeroes of that function inside the curve:

Suppose f is meromorphic (holomorphic except for on a set of isolated poles) on an open set which contains a closed curve \gamma and its interior, and that f has no zeroes or poles on \gamma. Then

\int _\gamma \frac{f'(z)}{f(z)}dz = 2 \pi i (Z - P),

where Z is the number of zeroes of f inside \gamma, and P the number of poles.



178: The Residue Formula

In complex analysis, the residue formula states that where f is a holomorphic function on an open set containing a circle C and its interior except for finitely many poles z_1, \dots, z_n in the interior of C,

\int _C f(z) dz = 2 \pi i \sum _{k = 1} ^n \text{res}_{z_k} (f).

That is, the integral of the circe is completely determined by the residues of f about the poles in its interior.

176: Removable Singularities vs. Poles

Let f: \Omega \rightarrow \mathbb{C} be holomorphic on \Omega except at isolated points where it’s undefined. One such isolated point z_0 is said to be removable if f can be extended to include z_0 such that the extension is holomorphic in a neighborhood of z_0. That is, if z_0 is “correctable”.

By contrast, an isolated undefined point is said to be a pole if not removable, but there’s a positive integer n such that z_0 is a removable singularity of the function

f(z)(z - z_0)^n.

In this case, n is called the order of the pole.

171: More General Cauchy Integral Theorem

Let \Omega \subseteq \mathbb{C} be open, and let \Gamma \subset \Omega be a cycle which is null homologous. (That is, \Gamma is a collection of closed curves the winding number of which is 0.) Then

\int _\Gamma f(z) dz = 0.

154: Morera’s Theorem

In complex analysis, Morera’s Theorem states that if D \subset \mathbb{C} is a region (open, path-connected) and f: D \rightarrow \mathbb{C} is continuous and has the property that

\int _\gamma f(z) dz = 0

for any closed curve \gamma \subset D, then f is holomorphic.

This is an immediate corollary to the observation in [153], since if F is a primitive of f on D then f itself is holomorphic on D.

153: Sufficient Condition for Existence of Primitive of Complex Function

If \Omega \subset \mathbb{C} is open and path-connected, and a continuous function

f: \Omega \rightarrow \mathbb{C}

has the property that

\int _\gamma f(z) dz = 0

for any closed curve \gamma \subset \Omega, then f has a primitive (antiderivative) on \Omega.

(The converse is obviously true, since if F is a primitive of f then for any closed curve \gamma which begins and ends at a point z, \int _\gamma f(z) dz = F(z) - F(z) = 0.)



147: Cauchy Integral Formula

Suppose \Omega \subset \mathbb{C} is open, f: \Omega \rightarrow \mathbb{C} be holomorphic, and C \subset \Omega be the boundary of a disk as a curve with positive (counterclockwise) orientation. Then where z belongs to the interior of this disk,

f(z) = \frac{1}{2 \pi i} \int _C \frac{f(w)}{w - z} dw.

That is, the value of f on the interior of the disk is completely determined by its values on the boundary of the disk. This is really cool!

145: Cauchy’s Theorem (Complex Analysis)

Let \Omega be an open subset of \mathbb{C}, \gamma \subset \Omega be a closed curve whose interior is contained in \Omega, and f: \Omega \rightarrow \mathbb{C} be holomorphic. Then Cauchy’s Theorem states that

\int _{\gamma} f(z) dz = 0.

143: Complex Integrals Over Curves With Identical Sets but Opposite Orientation

Given a curve \gamma, let \gamma ^{-} be the curve which covers the same subset of \mathbb{C} as \gamma, but has opposite orientation (goes in the other direction). Then

\int_{\gamma^{-}} f(z) dz = - \int_{\gamma} f(z) dz,

assuming the integral over \gamma exists in the first place. So reversing the direction of a curve just reverses the sign of an integral over that curve.

138: Projected Circles on the Reimann Sphere

Under the Reimann Sphere projection, circles on the sphere are mapped to “circles” in the extended complex plane \mathbb{C}^*.

The quotes are because every line in \mathbb{C}^* is mapped to by a circle on the Reimann Sphere passing through the north pole \mathcal{N}. So in this view, lines in \mathbb{C} are just circles in \mathbb{C}^*.