# Today I Learned

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

## 191: Stability, Instability of Numerical Integration, Differentiation

Numerical integration methods such as use of the composite Simpson’s rule, composite trapezoidal rule, etc., are stable. That is, their round-off error as a result of being performed with machine arithmetic does not depend on the number of sub-intervals the integrating interval is split up into.

By contrast, numerical differentiation methods such as the 2-point, 3-point, and 5-point formulas are unstable, because these formulas involve division by the width of the step-size $h$ used. So at a certain point as $h \rightarrow 0$, round-off error overtakes the increased accuracy of a small step-size and the approximations actually start to get worse.

## 188: Lagrange Interpolation

Given a real function $f$, a set of $n+1$ points $x_0, \dots, x_n$, and the values $f(x_0), \dots, f(x_n)$ of $f$ at those points, the Lagrange polynomial uses this information to give an $n$-th degree polynomial $P$ which approximates $f$. The formula for $P$ is

$P = L_0 f(x_0) + \cdots + L_n f(x_n)$,

where

$L_k = (\prod_{i \neq k} (x - x_i))/(\prod_{j \neq k} (x_k - x_j))$.

Example: where $n = 1$ and we are given $x_0, x_1, f(x_0), f(x_1)$, the Lagrange polynomial approximating $f$ is

$P = \frac{x - x_1}{x_0 - x_1} f(x_0) + \frac{x - x_0}{x_1 - x_0} f(x_1)$.

Examination of the polynomial should make it clear that it is, in a way, the simplest polynomial where $P(x_k) = f(x_k)$ at each $x_k$.

## 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

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