9 - Power Series 2.0 Flashcards
Uniform convergence of power series, Weierstrass test. Continuity, differentiation and integration using uniform convergence and application to power series. Uniqueness of power series, analytic functions.
What is the Cauchy-Hadamard Theorem?
R=1/(lim_(n->infinity)^_(|a_n|)^(1/n)).
What is term-wise differentiation?
Let f(z) = ∞∑(k=0) a_k * z^n be a power series with a radius of convergence R > 0. Then R is the radius of convergence for the power series
g(z) := ∞∑(k=1) k * a_k * z^(k-1) = ∞∑(k=0) (k+1) * a_k * z^k
F(z) := ∞∑(k=0) (1/(k+1)) * a_k * z^(k+1)
Moreover, F and f are differentiable with F’ = f and f’ = g.
How do we represent f by its Taylor series around any z_0 in B(0,R)?
Let f(z) = ∞∑(k=0) a_k * z^k be a power series with a radius of convergence R > 0. If z0 is in B(0,R), then
f(z) = ∞∑(k=0) 1/k! * f^(k) (z0) * (z-z0)^k
For all z such that |z - z0| < R - |z0|
What is the double series theorem?
A double sum is a series having terms depending on two indices:
∞∑(i,j) b_(i,j)
An infinite double series can be written in terms of a single series:
∞∑(k=0) * ∞∑(l=0) c_(k,l) = ∞∑(i=0) b_i
What is the uniqueness theorem for power series?
Let f(z) = ∞∑(k=0) a_k * z^k and g(z) = ∞∑(k=0) b_k * z^k be power series converging for |z| < R. Suppose that z_n /= 0 for all natural numbers n and that z_n –> 0. If f(z_n) = g(z_n) for all natural numbers n, then a_k = b_k for all natural numbers k.
What are analytic functions?
We say that f is analytic on D (domain) if for every x_0 in the domain there exists R > 0 and a sequence a_k such that
f(x) = ∞∑(k=0) a_k * (x - x0)^k
for all x in the set B(x_0, R). If the domain is the set of real numbers, we say f is real analytic. The same applies for the domain of complex numbers.
Are functions defined by a power series analytic?
Every function defined by a power series is analytic.
What is a path?
A path in a set D is a continuous function γ: [a,b] –> D. We write γ ϵ C([a,b], D)
What is a connected set?
An open set D is called (path) connected if for every pair x,y in D there exists a path γ ϵ C([0,1], D) with γ(0) = x and γ(1) = y.
What is the uniqueness theorem for analytic functions?
Let {D} (an improper subset of {R}) be connected and f,g: {D} –> {R} be analytic functions on {D}. Moreover, suppose that (x_n) is a sequence in {D} with x_n –> x_0, x_0 being in {D} and x_n /= x_0 for all n in {N}. If f(x_n) = g(x_n) for all n in {N}, then f(x) = g(x) for x in {D}.
What is a continuation argument?
Analytic continuation is a technique to extend the domain of the definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.
What are some operations of the exponential function?
The function exp: {C} –> {C} is analytic. Moreover,
- exp(0) = 1 and exp(1) = e
- exp(z+w) = exp(z) * exp(w) for all z,w in {C}
- exp(z) /= 0 and exp(-z) = 1/exp(z) for all z in {C}
- Conjugate of exp(z) = exp(conjugate of z) for all z in {C}
- |exp(it)| = 1 for all t in {R}
- d/dz exp(z) = exp(z) for all z in {C}
- exp: {R} –> (0, infinity) is strictly increasing and bijective
What are some operations of the real logarithm function?
The function log: (0, infinity) –> {R} is analytic. Moreover,
- log(1) = 0 and log(e) = 1
- log(x*y) = log(x) + log(y) for all x,y > 0.
- d/dx log(x) = 1/x for all x > 0
- log: (0, infinity) –> {R} is strictly increasing and bijective
What does the complex logarithm look like?
log(x) = log(x_0) + ∞∑(k=1) ((-1)^(k-1) / (k*x^k_0) * (x - x0)^k
What are some properties of the complex logarithm function?
There exists a unique function Log: {C} \ {0} –> {C} with the property that
- Log 1 = 0
- d/dz Log z = 1/z for all z in {C} \ (- infinity, 0]
- exp(Log z) = z for all z in {C} \ {0}.
More over, Log: {C} \ (- infinity, 0] –> {C} is analytic.
