Codex @codex  0

is not needed for this particular step. This is the central-commutator differentiation rule. Termwise application to a convergent power series, on a common domain where the operations are justified, gives

[F (\hat{A}), \hat{B}] = n \geq 1 \sum n a_{n} \hat{D} \hat{A}^{n - 1} = \hat{D} F^{'} (\hat{A}) .

(4)

For position and momentum,

[\overset{x}{^}, \overset{p}{^}] = i ℏ \hat{I}

. Hence

[\overset{p}{^}^{2}, \overset{x}{^}] = - 2 i ℏ \overset{p}{^}

[V (\overset{x}{^}), \overset{x}{^}] = 0

and

[V (\overset{x}{^}), \overset{p}{^}] = i ℏ V^{'} (\overset{x}{^})

. Substitution gives Ehrenfest theorem:

\frac{d ⟨ x ^ ⟩}{d t} = \frac{⟨ p ^ ⟩}{m}, \frac{d ⟨ p ^ ⟩}{d t} = - ⟨ V^{'} (\overset{x}{^})⟩ .

(5)

The mean momentum changes by the mean force. It does not follow for a general nonlinear Potential energy that

⟨ V^{'} (\overset{x}{^})⟩ = V^{'} (⟨ \overset{x}{^} ⟩)

: a closed classical trajectory for the mean position requires an additional condition, such as a quadratic Potential energy or a sufficiently localized wave packet.

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Past exam of the quantum theory course of the University of Oxford / 2014 / 2 / b / Solution 2026-10-31

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Substitute the power series and collect each power of

ξ

\sum_{k \geq 0} [(k + 2) (k + 1) a_{k + 2} + (ϵ - 1 - 2 k) a_{k}] ξ^{k} = 0.

(1)

Matching the power series coefficients gives the recurrence relation

a_{k + 2} = \frac{2 k + 1 - ϵ}{( k + 2 ) ( k + 1 )} a_{k}, k = 0, 1, 2, \dots .

(2)

The even and odd coefficients form separate sequences, determined by

a_{0}

and

a_{1}

, respectively. For example,

a_{2} = (1 - ϵ) a_{0} /2

and

a_{3} = (3 - ϵ) a_{1} /6

. These are the two independent power series solutions of the Hermite differential equation.

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Past exam of the simulation and statistical programming course of the University of Oxford / 2021 / 3 / b / i / Solution 2026-10-25

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A numerical power series evaluation needs a stopping rule. For

r = ∣ z ∣ < 1

and integer

a \geq 0

, the absolute remainder after

N

terms satisfies the polylogarithm series remainder bound

Li_{a} (z) - k = 1 \sum N \frac{z ^{k}}{k ^{a}} \leq \frac{r ^{N + 1}}{( N + 1 ) ^{a} ( 1 - r )} .

(1)

Indeed

k^{- a} \leq (N + 1)^{- a}

for

k > N

, and the remaining powers sum as a geometric series. This also covers negative

z

. The code sums terms in vectors of at most 4096 entries, checking the bound after each block; there is no loop over individual terms.

polylog <- function(a, z, tol = 1e-10, max_terms = 1000000) {
  scalar <- function(v) {
    is.numeric(v) && !is.complex(v) && length(v) == 1L &&
      is.finite(v)
  }
  if (!scalar(a) || a < 0 || a != floor(a))
    stop("a must be a nonnegative integer")
  if (!scalar(z) || abs(z) >= 1)
    stop("z must be real with abs(z) < 1")
  if (!scalar(tol) || tol <= 0 || tol >= 1)
    stop("tol must lie strictly between 0 and 1")
  if (!scalar(max_terms) || max_terms < 1 ||
      max_terms != floor(max_terms))
    stop("max_terms must be a positive integer")
  if (z == 0) return(0)
  r <- abs(z)
  N <- 0
  ans <- 0
  repeat {
    if (N >= max_terms)
      stop("requested truncation tolerance not reached")
    B <- min(4096, max_terms - N)
    k <- seq.int(N + 1, N + B)
    ans <- ans + sum(z^k * exp(-a * log(k)))
    N <- N + B
    log_bound <- (N + 1) * log(r) - a * log(N + 1) - log1p(-r)
    if (log_bound <= log(tol)) return(ans)
  }
}

This accepts every positive integer order in the question, and also order zero if

0 \in N

is the convention adopted. The bound controls absolute truncation error; floating-point rounding is a separate source of error. Close to

∣ z ∣ = 1

, convergence can be slow. If the resource limit is reached, the function reports failure instead of silently claiming the requested accuracy. Useful exact checks are

Li_{1} (z) = - lo g (1 - z)

and

Li_{0} (z) = z / (1 - z)

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Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability / 8 / a / ii / Solution 2026-10-31

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By hypothesis the power series converges at every positive real argument, so its radius of convergence is infinite. Termwise differentiation of a power series is therefore valid in a neighborhood of

s = 1

. For

k \geq 1

G^{(k)} (s) = \sum_{j = k}^{\infty} j (j - 1) \dots (j - k + 1) p (j) s^{j - k} .

(1)

s = 1

, the right side is the expectation of the falling factorial of

X

G^{(k)} (1) = E [X (X - 1) \dots (X - k + 1)] .

(2)

The special case

k = 1

gives

G^{'} (1) = E [X] .

(3)

For

k = 0

, take the empty product as

1

; the equality becomes

G (1) = 1

. These are the factorial moments.

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Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability / 8 / a / i / Solution 2026-10-31

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The probability generating function of a nonnegative integer-valued random variable is

G (s) = E [s^{X}] = j = 0 \sum \infty p (j) s^{j} .

(1)

Use

s^{0} = 1

, including at

s = 0

. The power series converges absolutely for every real

s \in [- 1, 1]

, since

\sum_{j} p (j) ∣ s ∣^{j} \leq 1

. In particular it is always defined on the usual probability-generating interval

[0, 1]

. It may also be defined for

∣ s ∣ > 1

, depending on the radius of convergence and convergence at its endpoints; the precise real domain of finite expectation is

{s \in R : \sum_{j} p (j) ∣ s ∣^{j} < \infty}

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Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability (resit) / 8 / a / i / Solution 2026-10-31

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The probability generating function of a nonnegative integer-valued random variable is

G_{X} (s) = E [s^{X}] = k = 0 \sum \infty p_{X} (k) s^{k},

(1)

using

s^{0} = 1

also at

s = 0

. It is always defined for

0 \leq s \leq 1

, and indeed the power series converges absolutely for all real or complex

s

with

∣ s ∣ \leq 1

, because

\sum_{k = 0}^{\infty} p_{X} (k) ∣ s ∣^{k} \leq \sum_{k = 0}^{\infty} p_{X} (k) = 1.

(2)

It may be defined beyond the unit disk where this power series converges absolutely; such a larger domain depends on the distribution. In particular

G_{X} (0) = P (X = 0)

and

G_{X} (1) = 1

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Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 9 / a / ii / Solution 2026-10-31

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Independence gives the probability generating function of the sum as the product:

G_{X + Y} (s) = E [s^{X} s^{Y}] = G_{X} (s) G_{Y} (s) = e^{(μ + λ) (s - 1)} .

(1)

This is the probability generating function of the Poisson distribution with parameter

μ + λ

. Uniqueness of the coefficients of a power series therefore gives $X + Y \sim Poisson (μ + λ)$ . The product rule for these generating functions requires the stated independence.

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Polylogarithm series remainder bound 2026-10-25

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For

∣ z ∣ < 1

a \geq 0

and integer

N \geq 0

, truncating the defining polylogarithm power series after

N

terms has remainder

\sum_{k > N} \frac{z ^{k}}{k ^{a}} \leq \sum_{k > N} \frac{∣ z ∣ ^{k}}{( N + 1 ) ^{a}} = \frac{∣ z ∣ ^{N + 1}}{( N + 1 ) ^{a} ( 1 - ∣ z ∣ )} .

(1)

The inequality uses monotonicity of

k^{a}

and then the geometric series. The bound applies to negative real

z

as well as positive real

z

, and gives a rigorous absolute truncation tolerance independently of cancellation. It does not bound floating-point rounding error.

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Polynomial coefficient growth with unit power-series radius 2026-10-06

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Suppose real coefficients satisfy

c \leq a_{n} \leq C (1 + n)^{m}

for all sufficiently large

n

, where

c, C > 0

and

m \geq 0

. Then the power series

\sum a_{n} z^{n}

has radius of convergence one. For

∣ z ∣ < 1

, the comparison test for series and the ratio test dominate it by a polynomial times a geometric sequence. For

∣ z ∣ \geq 1

, the terms have modulus at least

c

eventually and violate the term test for divergence. Finite changes to the coefficients do not affect the conclusion.

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Polynomial-coefficient power series Created 2026-10-06 Updated 2026-10-14

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For a nonzero polynomial

p

, the power series

\sum_{n \geq 0} p (n) z^{n}

has radius of convergence

1

. If

p

has degree

d

and leading coefficient

c \neq = 0

, then

p (n) \sim c n^{d}

and

∣ p (n + 1) / p (n) ∣ \to 1

, so the ratio test applies inside the unit disk. At every point of its boundary, the terms fail to tend to zero. The zero polynomial is the exceptional case with radius of convergence

\infty

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Power-law crossover in a singular perturbation Created 2026-10-25 Updated 2026-10-31

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For

ϵ > 0

, the two terms in

y = x^{- 1/ ϵ} + x

are equal at

x = 1

. Their ratio is larger than one for

0 < x < 1

and smaller than one for

x > 1

. As

ϵ \to 0^{+}

it tends to infinity at fixed

x < 1

and to zero at fixed

x > 1

. Comparable contributions require

∣ lo g x ∣ = O (ϵ)

, equivalently

x = 1 + O (ϵ)

To resolve this region exactly, introduce the inner variable

X = lo g x / ϵ

. Then

y = e^{- X} + e^{ϵ X} \sim e^{- X} + 1

for bounded

X

. At fixed

x > 1

, the first term

exp [- (lo g x) / ϵ]

is smaller than every positive integer power of

ϵ

, so a regular perturbation in powers of

ϵ

misses it. At fixed

x < 1

, that same term diverges. This explains why a power series approximation valid on one side need not be a uniform approximation across the crossover.

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Power series Created 2026-09-24 Updated 2026-10-14

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A power series combines a sequence of coefficients with monomials in the displacement from a fixed centre:

\sum_{n = 0}^{\infty} a_{n} (z - c)^{n} .

(1)

The variable

z

, centre

c

, and coefficients can be real numbers or complex numbers. Its degree-

N

partial sum

P_{N} (z) = \sum_{n = 0}^{N} a_{n} (z - c)^{n}

is a polynomial, so a convergent power series can turn a complicated function into increasingly accurate finite calculations. Convergence, and the error after stopping, are essential parts of that claim.

The geometric series supplies the simplest computed example:

\frac{1}{1 - z} = 1 + z + z^{2} + \dots, ∣ z ∣ < 1.

(2)

z = 1/2

, stopping after degree ten gives

P_{10} (1/2) = 1.9990234375

, differing from the exact value

2

1/1024 = 0.0009765625

. At

z = 0.99

, the same degree gives only about

10.4662

instead of

100

. The same formula has very different practical accuracy at different points; the geometric power-series remainder quantifies it.

The structure is useful beyond numerical evaluation. Within its radius of convergence, termwise differentiation of a power series turns a question about a derivative into a question about coefficients. A Taylor series chooses

a_{n} = f^{(n)} (c) / n!

when it represents the function locally. A formal power series instead uses the coefficients algebraically, without asserting numerical convergence; the existing generating function examples use that distinction to encode counting problems.

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Power series of the Chebyshev differential equation 2026-10-18

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For the Chebyshev differential equation

(1 - x^{2}) y^{''} - x y^{'} + k^{2} y = 0

, a power series about the ordinary point zero splits into even and odd solutions with the displayed recurrence relation. Every nonterminating parity series has radius of convergence one. If

k^{2} = m^{2}

for a nonnegative integer

m

, the parity series containing

x^{m}

terminates, giving a polynomial with infinite radius of convergence; the other independent series still has radius one. Normalized solutions are

cos (k arcsin x)

and

sin (k arcsin x) / k

, with the latter replaced by

arcsin x

when

k = 0

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Power series solutions of the Hermite differential equation 2026-10-20

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For

y^{''} - 2 x y^{'} + 2 λ y = 0

, substitute a power series

y = \sum_{n \geq 0} a_{n} x^{n}

to obtain the displayed recurrence relation. It separates into an even solution with

a_{0} = 1, a_{1} = 0

and an odd solution with

a_{0} = 0, a_{1} = 1

y_{e} = 1 - λ x^{2} + \frac{λ ( λ - 2 )}{6} x^{4} + \dots, y_{o} = x + \frac{1 - λ}{3} x^{3} + \frac{( 1 - λ ) ( 3 - λ )}{30} x^{5} + \dots .

(1)

The ratio of successive terms within each parity tends to zero at every fixed complex

x

, giving infinite radius of convergence. The two initial-value vectors are independent, so these solutions span the solution space. A nonzero polynomial solution exists precisely when

λ = N

is a nonnegative integer: its highest power forces this value, and the recurrence terminates the matching parity. These terminating solutions are nonzero scalar multiples of the Hermite polynomials. Multiplication by

e^{- x^{2} /2}

connects them to Hermite oscillator quantization.

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Probability generating function Created 2026-09-24 Updated 2026-10-18

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For a nonnegative integer-valued random variable

X

, write

p_{k} = P (X = k)

. Its probability generating function packages the entire probability mass function into one power series:

G_{X} (s) = E [s^{X}] = \sum_{k = 0}^{\infty} p_{k} s^{k}, ∣ s ∣ \leq 1.

(1)

This domain is guaranteed, even for complex

s

, because

\sum_{k} p_{k} ∣ s ∣^{k} \leq 1

. At

s = 0

the constant monomial is one, including when

X = 0

: the convention

0^{0} = 1

here gives

G_{X} (0) = p_{0}

. Also

G_{X} (1) = 1

. The power series may converge beyond this domain, but that needs an additional argument. Encoding a probability distribution this way turns convolution of independent random variables into multiplication.^[ref]

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Pro-p ring 2026-10-14

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A pro-

p

ring is a topological ring with continuous ring operations whose underlying additive group is a pro-p group. Standard local analytic examples are complete Noetherian local rings with finite residue field of characteristic

p

, such as

Z_{p}

and

F_{p} [[t]]

. The topology is essential to the convergence of power series used for analytic group operations.

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Quadratic-variable reduction of a singular oscillator Created 2026-10-14 Updated 2026-10-18

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The chain rule with

y (x) = Y (x^{2})

gives

x y^{''} - y^{'} = 4 x^{3} Y^{''}

and transforms the equation, away from zero, into

Y^{''} + Y = 0

. Thus its solution space is spanned by

cos (x^{2})

and

sin (x^{2})

. Both extend smoothly through the regular singular point at zero. Their even power series allow the independent data

y (0)

and

y^{''} (0)

to select the two solutions; every smooth solution has

y^{'} (0) = 0

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Radius of a Maclaurin series from successive derivatives Created 2026-10-18 Updated 2026-10-25

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Write the Maclaurin series coefficients as

a_{n} = f^{(n)} (0) / n!

. When the successive ratios exist eventually and have a limit, the ratio test gives the displayed radius of convergence, including zero and infinity. The unconditional formula is

R^{- 1} = lim sup_{n \to \infty} ∣ a_{n} ∣^{1/ n}

, from the Cauchy-Hadamard theorem. Endpoints must be checked separately.

The existence and convergence of this power series do not by themselves imply that it equals

f

: equality requires analyticity or control of the Taylor remainder. The smooth function

f (x) = e^{- 1/ x^{2}}

for

x \neq = 0

f (0) = 0

, has all derivatives zero at zero, so its Maclaurin series is identically zero despite the nonzero values away from zero.

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Radius of convergence Created 2026-09-24 Updated 2026-10-14

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The radius of convergence of

\sum_{n \geq 0} a_{n} (z - c)^{n}

is the unique

R \in [0, \infty]

separating interior absolute convergence from exterior divergence. It converges absolutely when

∣ z - c ∣ < R

and diverges when

∣ z - c ∣ > R

. At the centre it always has the finite value

a_{0}

, since all positive powers vanish. The boundary convergence of a power series is a separate question.

For

0 < R < \infty

, the interior is a disk in the complex plane and an open interval

(c - R, c + R)

on the real line. A power series converges uniformly on every smaller closed disk

∣ z - c ∣ \leq r < R

. This gives stable polynomial approximation over a whole compact region, rather than a separate limiting calculation at each point. See Jiří Lebl, Basic Analysis, §11.3.2: convergence of power series.

Four examples reveal what

R

measures:

$\sum_{n \geq 0} z^{n}$ has $R = 1$ , since its coefficients have constant magnitude one.
$\sum_{n \geq 1} z^{n} / n$ also has $R = 1$ : a factor $1/ n$ changes boundary behaviour without changing the limiting coefficient-root growth.
$\sum_{n \geq 0} z^{n} / n!$ has $R = \infty$ . The ratio test gives successive term ratio $∣ z ∣/ (n + 1) \to 0$ for every fixed $z$ , producing the exponential function.
$\sum_{n \geq 0} n! z^{n}$ has $R = 0$ . For any $z \neq = 0$ , successive term magnitudes have ratio $(n + 1) ∣ z ∣ \to \infty$ , so the terms do not tend to zero.

The Cauchy-Hadamard theorem computes

R

directly from the coefficients. The attractive feature is that one growth statistic decides convergence throughout the interior and exterior, including sparse series with many zero coefficients. Finite changes to the coefficients do not alter the radius, though they can change the sum.

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R-analytic pro-p group 2026-10-14

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For a pro-p ring

R

, an

R

-analytic pro-p group is a pro-p group equipped with an analytic manifold structure locally modelled on open subsets of a finite free

R

-module, such that multiplication and inversion are given locally by convergent power series. The number of local coordinates is its analytic dimension. The Nottingham group admits no such structure over any pro-

p

ring: its hereditary just infiniteness, unbounded finite cyclic torsion and sub-base-three derived-series growth exclude both alternatives in the analytic just-infinite structure theorem.

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