Today I Learned

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

Category Archives: analysis

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.

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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.