nyuuzyou/wikis huggingface dataset:
Reversed Contour Reverses Orientation/Corollary

== Corollary ==
Let $C$ be a simple closed contour in the complex plane $\C$.
Let $-C$ be the reversed contour of $C$.
If $C$ is positively oriented, then $-C$ is negatively oriented.
If $C$ is negatively oriented, then $-C$ is positively oriented.
== Proof ==
From Orientation of Simple Closed Contour is with Respect to Interior, it follows that:
:$C$ is positively oriented, $C$ is positively oriented with respect to $\Int C$.
:$C$ is negatively oriented, $C$ is negatively oriented with respect to $\Int C$.
where $\Int C$ denotes the interior of $C$.
The claims of the corollary now follow from the main theorem Reversed Contour Reverses Orientation.
Category:Orientation of Complex Contour
Reversed Contour is Contour

== Theorem ==
Let $\R^n$ be a real cartesian space of $n$ dimensions.
Let $C$ be a contour in $\R^n$ that is defined as a concatenation of a finite sequence $C_1, \ldots, C_n$ of directed smooth curves in $\R^n$.
Then the finite sequence of reversed directed smooth curves:
:$-C_n, -C_{n - 1}, \ldots, -C_1$
defines a contour that is independent of the parameterization of $C_1, \ldots, C_n$.
== Proof ==
Let $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$.
From Reversed Directed Smooth Curve is Directed Smooth Curve, it follows that $-C_i$ is independent of the paraterization $\gamma_i$ of $C_i$.
We now prove that the end point of $-C_i$ is equal to the start point of $-C_{i - 1}$ for all $i \in \set {2, \ldots, n}$.
By definition of reversed directed smooth curve, $-C_i$ is parameterized by $\rho_i: \closedint {a_i} {b_i} \to \C$.
Here, $\map {\rho_i} t = \map {\gamma_i} {a_i + b_i - t}$.
From Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints, it follows that the endpoints $\map {\rho_i} {a_i}$ and $\map {\rho_i} {b_i}$ are independent of the parameterization $\rho_i$.
Then:
By definition, it follows that $-C_n, \ldots, -C_1$ can be concatenated to form a contour.
== Sources ==
* $\S 2.2$
Category:Contour Integrals
Reversed Directed Smooth Curve is Directed Smooth Curve

== Theorem ==
Let $C$ be a directed smooth curve in $\C$.
Let $C$ be parameterized by the smooth path $\gamma: \left[{a \,.\,.\, b}\right] \to \C$.
Define $\psi: \left[{a \,.\,.\, b}\right] \to \left[{a \,.\,.\, b}\right]$ by $\psi \left({t}\right) = a + b - t$.
Define $\rho: \left[{a \,.\,.\, b}\right] \to \C$ by $\rho = \gamma \circ \psi$.
Then $\rho$ is a smooth path which parameterizes a directed smooth curve $-C$.
The directed smooth curve $-C$ is independent of the parameterization $\gamma$.
== Proof ==
First, we prove that $\rho$ is a smooth path:
As $\gamma'$ is continuously differentiable, and $\psi$ is continuous, it follows that $\rho'$ is continuous.
Then $\rho$ is a parameterization of a directed smooth curve $-C$. 
Next, we prove that the definition of $-C$ is independent of the parameterization $\gamma$.
Suppose $\sigma$ is another parameterization of $C$:
: $\sigma = \gamma \circ \phi$
Here $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ is a bijective differentiable strictly increasing function.
Define $\tilde \psi: \left[{c \,.\,.\, d}\right] \to \left[{c \,.\,.\, d}\right]$ by $\tilde \psi \left({t}\right) = c + d - t$.
Define $\tilde \rho: \left[{c \,.\,.\, d}\right] \to \C$ by $\tilde \rho = \sigma \circ \tilde{\psi}$.
We now prove that $\rho$ and $\tilde \rho$ both are parameterizations of the same directed smooth curve $-C$.
Both $\psi$ and $\tilde \psi$ are bijective with $\psi^{-1} = \psi$ and $\tilde \psi^{-1} = \tilde \psi$.
Define $\tilde \phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ by $\tilde \phi = \psi^{-1} \circ \phi \circ \tilde \psi$.
From Composite of Bijections is Bijection, it follows that $\tilde \phi$ is bijective.
From Derivative of Composite Function, it follows that $\tilde \phi$ is differentiable with
From Derivative of Monotone Function, it follows that $\tilde \phi$ is strictly increasing.
As:
:$\rho \circ \tilde \phi = \gamma \circ \psi \circ \psi^{-1} \circ \phi \circ \tilde \psi = \gamma \circ \phi \circ \tilde \psi = \sigma \circ \tilde \phi = \tilde \rho$
it follows that $\rho$ and $\tilde{\rho}$ are parameterizations of $-C$.
== Sources ==
* $\S 2.2$
Category:Directed Smooth Curves (Complex Plane)
Reversion of Power Series

== Theorem ==
Let $\ds y = \sum_{n \mathop = 1}^\infty c_n x^n$ be a power series.
Then:
:$\ds x = \sum_{n \mathop = 1}^\infty C_n y^n$
is also a power series, where:
A derivation of the explicit formula for the $n$th term is given by:
:$\ds C_n = \dfrac 1 {n {c_1}^n} \sum_{s, t, u, \cdots} \paren {-1}^{s + t + u + \cdots} \dfrac {n \paren {n + 1} \cdots \paren {n - 1 + s + t + u + \cdots} } {s! \, t! \, u! \ldots} \paren {\dfrac {c_2} {c_1} }^s \paren {\dfrac {c_3} {c_1} }^t \cdots$
where $ s + 2 t + 3 u + \cdots = n - 1$.
== Proof ==
Plugging:
:$\ds x = \sum_{n \mathop = 1}^\infty C_n y^n$
into:
:$\ds y = \sum_{n \mathop = 1}^\infty c_n x^n$
the following equation is obtained
:$y = c_1 C_1 y + \paren {c_2 C_1^2 + c_1 C_2} y^2 + \paren {c_3 C_1^3 + 2 c_2 C_1 C_2 + c_1 C_3} y^3 + \paren {3 c_3 C_1^2 C_2 + c_2 C_2^2 + c_2 C_1 C_3} + \cdots$
Equating coefficients then gives the formula.
== Sources ==
* : Part $1$
* : $\S 20$: Reversion of Power Series: $20.52 - 59$
* : $\S 22$: Taylor Series: Reversion of Power Series: $22.52 - 59.$
Category:Power Series
Rhind Papyrus 40

== Problem ==
:A hundred loaves to five men, one-seventh of the three first men to the two last.
== Solution ==
This means:
:Divide $100$ loaves between $5$ men so that:
::the shares are in arithmetic progression
::the sum of the two smaller shares is $\dfrac 1 7$ of the sum of the $3$ greatest.
Thus, let:
:$a$ be the smallest share, that is, the initial term
:$d$ be the common difference.
We have:
So the shares are:
== Sources ==
* : Problem $30$
* : Problem $30$
* : Sharing the Loaves: $10$
Category:Rhind Papyrus
Riccati Equation/y' = (y over x) + x^3 y^2 - x^5

== Theorem ==
The Riccati equation:
:$(1): \quad y' = \dfrac y x + x^3 y^2 - x^5$
has the general solution:
:$C \exp \dfrac {2 x^5} 5 = \dfrac {y - x} {y + x}$
== Proof ==
It can be seen by inspection that:
:$\map {y_1} x = x$
is a particular solution to $(1)$.
Thus from General Solution of Riccati Equation from Particular Solution:
:$y = x + \map z x$
where:
:$z' - \paren {\dfrac 1 x + 2 x^4} z = x^3 z^2$
From Bernoulli's Equation: $y' - \paren {\dfrac 1 x + 2 x^4} y = x^3 y^2$:
:$z = \dfrac {2 x} {C \map \exp {-\dfrac {2 x^5} 5} - 1}$
Thus:
== Sources ==
* : Miscellaneous Problems for Chapter $2$: Problem $28$
Category:Examples of Riccati Equation
Richard's Paradox

== Paradox ==
Certain expressions of natural language define real numbers unambiguously, while other expressions of natural language do not.
For example:
:The real number whose integer part is $17$ and whose $n$th decimal place is $0$ if $n$ is even and $1$ if $n$ is odd
defines the real number $17.1010101 \ldots = \dfrac {1693} {99}$.
However, the phrase:
:the capital of France
does not define a real number. 
Neither does the phrase:
:the smallest positive integer not definable in under sixty letters
because the expression does not define a number unambiguously (see Berry's Paradox).
Thus there exists an infinite list of English phrases of finite length that define real numbers unambiguously.
Let this infinite list of phrases be ordered linearly:
:$(1): \quad$ by increasing length
:$(2): \quad$ then order all phrases of equal length in lexicographic order, for example, using the ASCII code, such that the phrases can only contain codes $32$ to $126$).
This ordering is declared to be the canonical form of the sequence of the real numbers.
Thus we have an infinite sequence of the corresponding real numbers:
:$r_1, r_2, \ldots$
Now define a new real number $r$ as follows.
:The integer part of $r$ is $0$.
:The $n$th decimal place of $r$ is:
::$1$ if the $n$th decimal place of $r_n$ is not $1$
::$2$ if the $n$th decimal place of $r_n$ is $1$.
The preceding text on this page is an expression in the English language that unambiguously defines a real number $r$.
But because of the way $r$ was constructed, it cannot equal any of the $r_n$.
Thus, despite being unambiguously defined, it is at the same time undefinable.
This contradiction is known as Richard's paradox.
== Resolution ==
This is an antinomy.
The definition above of $r$ as refers to the definability itself, in natural language, of a real number.
If it were possible to determine which expressions in English actually do define a real number, and which do not, then the paradox would go through.
Thus the resolution of Richard's paradox is that there exists no way to unambiguously determine exactly which English sentences are definitions of real numbers.
That is, there exists no way to describe in a finite number of words how to tell whether an arbitrary English expression is or is not a definition of a real number.
Hence the root of the contradiction is the assumption that it is possible to unambiguously define every real number using a finite length expression in natural language.
== Also see ==
* Berry's Paradox
== Historical Note ==
== Sources ==
* 
* 
* 
Category:Antinomies
Category:Semantic Paradoxes
Richert's Theorem

== Theorem ==
Let $S = \set {s_1, s_2, \dots}$ be an infinite set of (strictly) positive integers, with the property:
:$s_n < s_{n + 1}$ for every $n \in \N$
Suppose there exists some integers $N, k$ such that every integer $n$ with $N < n \le N + s_{k + 1}$:
:$n$ can be expressed as a sum of distinct elements in $\set {s_1, s_2, \dots, s_k}$
:$s_{i + 1} \le 2 s_i$ for every $i \ge k$
Then for any $n > N$, $n$ can be expressed as a sum of distinct elements in $S$.
== Proof ==
We prove this using First Principle of Mathematical Induction.
Let $\map P n$ be the proposition:
:For every integer $m$ with $N < m \le N + s_{n + 1}$:
::$m$ can be expressed as a sum of distinct elements in $\set {s_1, s_2, \dots, s_n}$.
=== Basis for the induction ===
From our assumption above, $\map P k$ is true.
This is the basis for the induction.
=== Induction Hypothesis ===
Suppose for some $n > k$, $\map P {n - 1}$ is true.
That is, for every integer $m$ with $N < m \le N + s_n$:
:$m$ can be expressed as a sum of distinct elements in $\set {s_1, s_2, \dots, s_{n - 1}}$.
This is the induction hypothesis.
We need to show that $\map P n$ is true.
That is, for every integer $m$ with $N < m \le N + s_{n + 1}$:
:$m$ can be expressed as a sum of distinct elements in $\set {s_1, s_2, \dots, s_n}$.
=== Induction Step ===
This is the induction step:
By the induction hypothesis, for every integer $m$ with $N < m \le N + s_n$:
:$m$ can be expressed as a sum of distinct elements in $\set {s_1, s_2, \dots, s_{n - 1}}$.
Hence we only need to consider $N + s_n < m \le N + s_{n + 1}$.
For $m$ in this range:
:$N < m - s_n \le N + s_{n + 1} - s_n \le N + s_n$
By the induction hypothesis, $m - s_n$ can be expressed as a sum of distinct elements in $\set {s_1, s_2, \dots, s_{n - 1}}$.
Hence $m$ can be expressed as a sum of distinct elements in $\set {s_1, s_2, \dots, s_{n - 1}, s_n}$.
Hence $\map P n$ is true.
By the First Principle of Mathematical Induction, $\map P n$ is true for all $n \ge k$.
Since $1 \le s_n < s_{n + 1}$ for every $n \in N$:
:$s_n \ge n$ for every $n \in \N$
Let $K > N$.
Then:
By the result above, since $K + k \ge k$:
:$K$ can be expressed as the sum of distinct elements in $\set {s_1, s_2, \dots, s_{K + k}}$.
Thus every number greater than $N$ can be expressed as the sum of distinct elements in $S$.
Category:Number Theory
Riemann's Rearrangement Theorem

== Theorem ==
Let $S$ be a real infinite series which is conditionally convergent.
Then its terms can be arranged in a permutation so that the new series converges to any given value, or diverges.
== Proof ==
== Also known as ==
The Riemann's rearrangement theorem is also known as the Riemann series theorem.
== Historical Note ==
== Sources ==
* : Chapter $\text {A}.32$: Riemann ()
Category:Series
Riemann-Christoffel Tensor in Two Dimensions is Gaussian Curvature

== Theorem ==
Let $M$ be a Riemannian manifold of dimension $2$.
Then the Riemann-Christoffel tensor on $M$ reduces to the Gaussian curvature on $M$.
== Proof ==
== Historical Note ==
== Sources ==
* : Chapter $\text {A}.32$: Riemann ()
Category:Riemannian Geometry
Riemann-Christoffel Tensor in Two Dimensions is Gaussian Curvature/Historical Note

== Historical Note on Riemann-Christoffel Tensor in Two Dimensions is Gaussian Curvature ==
 demonstrated that the Riemann-Christoffel tensor in a Riemannian manifold of $2$ dimensions is the same thing as the Gaussian curvature in a posthumous paper on heat conduction.
== Sources ==
* : Chapter $\text {A}.32$: Riemann ()
Category:Historical Notes
Riemann-Hurwitz Formula

== Introduction ==
For a compact, connected, orientable surface $S$, the Euler characteristic $\chi(S)$ is
:$\chi(S)=2-2g$,
where g is the genus. This follows, as the Betti numbers are $1, 2g, 1, 0, 0, \dots$.
For the case of an (unramified) covering map of surfaces
:$\pi\colon S' \to S$
that is surjective and of degree $N$, we have the formula
:$\chi(S') = N\cdot\chi(S).$
That is because each simplex of $S$ should be covered by exactly $N$ in $S'$, at least if we use a fine enough triangulation of $S$, as we are entitled to do since the Euler characteristic is a topological invariant.
What the Riemann–Hurwitz formula does is to add in a correction to allow for ramification (sheets coming together):
==Theorem==
Now assume that $S$ and $S'$ are Riemann surfaces, and that the map $\pi$ is complex analytic.
The map $\pi$ is said to be ramified at a point $P$ in $S'$ if there exist analytic coordinates near $P$ and $π(P)$ such that $π$ takes the form $π(z) = z^n$, and $n>1$.
An equivalent way of thinking about this is that there exists a small neighborhood $U$ of $P$ such that $π(P)$ has exactly one preimage in $U$, but the image of any other point in $U$ has exactly $n$ preimages in $U$.
The number $n$ is called the ramification index at $P$ and is denoted by $e_P$. 
In calculating the Euler characteristic of $S'$ we notice the loss of $e_P-1$ copies of $P$ above $π(P)$ (that is, in the inverse image of $π(P)$). Now let us choose triangulations of $S$ and $S'$ with vertices at the branch and ramification points, respectively, and use these to compute the Euler characteristics. Then $S'$ will have the same number of $d$-dimensional faces for $d$ different from zero, but fewer than expected vertices. Therefore, we find a "corrected" formula
:$\chi(S') = N\cdot\chi(S) - \sum_{P\in S'} (e_P -1) $
or as it is also commonly written, using that $\chi(X) = 2 - 2g(X)$ and multiplying through by $-1$:
:$2g(S')-2 = N\cdot(2g(S)-2) +\sum_{P\in S'} (e_P -1) $
(all but finitely many $P$ have $e_P$ = 1, so this is quite safe).
==Proof==
==Other form==
:$\chi(S')- r = N \cdot (\chi(S) - b) $
where $r$ is the number points in $S'$ at which the cover has nontrivial ramification (ramification points) 
and $b$ is the number of points in $S$ that are images of such points (Branch points).
Indeed, to obtain this formula, remove disjoint disc neighborhoods of the branch points from $S$ and disjoint disc neighborhoods of the ramification points in $S' $ so that the restriction of $ \pi $ is a covering. Then apply the general degree formula to the restriction, use the fact that the Euler characteristic of the disc equals 1,
and use the additivity of the Euler characteristic under connected sums.
== Also known as ==
== Sources ==
* : $\S \text {IV}.2$
* 
Category:Riemann-Hurwitz Formula
Category:Algebraic Topology
Category:Algebraic Curves
Category:Riemann Surfaces
Category:Hurwitz's Theorem
Riemann-Lebesgue Lemma

== Lemma ==
Let $f \in L^1$.
Then:
:$\ds \lim_{n \mathop \to \infty} \int \map f x e^{i n x} \rd x = 0$
That is, the Fourier transform of an $L^1$ function vanishes at infinity.
== Proof ==
First suppose that $\map f x = \map {\chi_{\tuple {a, b} } } x$, the characteristic function of an open interval.
Then:
:$\ds \int \map f x e^{i n x} \rd x = \int_a^b e^{i n x} \rd x = \frac {e^{i n b} - e^{i n a} } {i n} \to 0$ as $n \to \infty$
By Sum Rule for Real Sequences, the same holds for an arbitrary simple function.
That is, for any function $f$ of the form:
:$\ds f = \sum_{n \mathop = 1}^N c_n \chi_{\tuple {a_i, b_i} }, \qquad c_n \in \R, \qquad a_i \le b_i \in \R$
we have that:
:$\ds \lim_{n \mathop \to \infty} \ds \int \map f x e^{i n x} \rd x = 0$
Finally, let $f \in L^1$ be arbitrary.
Let $\epsilon \in \R_{>0}$ be fixed.
From Space of Simple P-Integrable Functions is Everywhere Dense in Lebesgue Space, there exists a simple function $g$ such that:
:$\ds \int \size {f - g} < \epsilon$
By our previous argument and the definition of a limit of a complex function, there exists $N \in \N$ such that for all $n > N$:
:$\ds \size {\int \map g x e^{i n x} \rd x} < \epsilon$
By Integral of Integrable Function is Additive:
:$\ds \int \map f x e^{i n x} \rd x = \int \paren {\map f x - \map g x} e^{i n x} \rd x + \int \map g x e^{i n x} \rd x$
By the Triangle Inequality, Absolute Value of Definite Integral, Complex Modulus of Product of Complex Numbers and Euler's Formula:
:$\ds \size {\int \map f x e^{i n x} \rd x} \le \int \size {\map f x - \map g x} \rd x + \size {\int \map g x e^{i n x} \rd x}$
For all $n > N$, the is bounded by $2 \epsilon$ by our previous arguments.
Since $\epsilon$ was arbitrary, this establishes:
:$\ds \lim_{n \mathop \to \infty} \int \map f x e^{i n x} \rd x = 0$
for all $f \in L^1$.
Category:Fourier Analysis
Riemann-Roch Theorem/Historical Note

== Historical Note on Riemann-Roch Theorem ==
The Riemann-Roch theorem was stated and proved in the mid-$19$th century by and his student at the time .
 supposedly achieved his version by considering configurations of closed spaces and imagining experiments performed on them.
It has been used in recent times in the study of codes.
== Sources ==
* : Chapter $\text {A}.32$: Riemann ()
* 
* 
Category:Historical Notes
Riemann-Roch Theorem/Motivation

== Motivation for Riemann-Roch Theorem ==
The Riemann-Roch Theorem gives a formula which is used to calculate the exact number of independent holomorphic functions that can be defined on a given Riemann surface.
This formula involves the number of independent differential forms of special kinds.
Some of these are equal to topological invariants of the Riemann surface.
These are, in general, easy to calculate.
== Historical Note ==
== Sources ==
* 
* 
Category:Riemann-Roch Theorem
Riemann-Stieltjes Integral by Norm of Subdivision

== Theorem ==
Let $f, \alpha$ be real functions that are bounded on $\closedint a b$.
Suppose there exists some $A \in \R$ where, for every $\epsilon > 0$, there exists some $\delta_\epsilon > 0$ such that:
:For every finite subdivision $P$ of $\closedint a b$, if the norm $\norm P < \delta_\epsilon$, then:
::For every Riemann-Stieltjes sum $\map S {P, f, \alpha}$:
:::$\size {\map S {P, f, \alpha} - A} < \epsilon$
Then, $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a b$ and:
:$\ds \int_a^b f \rd \alpha = A$
=== Corollary ===
== Proof ==
Let $\epsilon > 0$ be arbitrary.
By hypothesis, let $\delta_\epsilon > 0$ be such that, for every subdivision $P$ with $\norm P < \delta_\epsilon$:
:$\size {\map S {P, f, \alpha} - A} < \epsilon$
By Existence of Subdivision with Small Norm, let $P_\epsilon$ be a subdivision of $\closedint a b$ such that:
:$\norm {P_\epsilon} < \delta_\epsilon$
Let $P$ be a subdivision that is finer than $P_\epsilon$.
Then:
Thus, by hypothesis:
:$\size {\map S {P, f, \alpha} - A} < \epsilon$
But, since $P$ finer than $P_\epsilon$ and $\epsilon > 0$ were arbitrary:
:$\ds \int_a^b f \rd \alpha = A$
by definition of the Riemann-Stieltjes integral.
== Sources ==
* : Chapter $7$ The Riemann-Stieltjes Integral: Exercise $7.3$
Category:Riemann-Stieltjes Integral
Riemann-Stieltjes Integral by Norm of Subdivision/Riemann Integral

== Theorem ==
Let $f$ be a real function that is bounded on $\closedint a b$.
Suppose $f$ is Riemann integrable on $\closedint a b$.
Let $\iota$ be the identity mapping on $\closedint a b$.
Then, $f$ is Riemann-Stieltjes integrable with respect to $\iota$ on $\closedint a b$ and:
:$\ds \int_a^b f \rd \iota = \int_a^b \map f x \rd x$
where the integral on the right is understood to denote the Riemann integral.
== Proof ==
Let $\epsilon > 0$ be arbitrary.
By definition of the Riemann integral, there exists some $\delta_\epsilon$ such that:
:For every finite subdivision $\Delta = \set {x_0, \dotsc, x_n}$ of $\closedint a b$, and every $C = \paren {c_i}_{1 \mathop \le i \mathop \le n}$ such that $c_i \in \closedint {x_{i - 1}} {x_i}$:
::If the norm $\norm \Delta < \delta_\epsilon$, then $\ds \size {\map S {f; \Delta, C} - \int_a^b \map f x \rd x} < \epsilon$
:where $\map S {f; \Delta, C}$ denotes a Riemann sum of $f$ for the subdivision $\Delta$.
Let $P = \set {x_0, \dotsc, x_n}$ be a finite subdivision of $\closedint a b$ such that $\norm P < \delta_\epsilon$.
Then:
As $P$ with $\norm P < \delta_\epsilon$ and $\epsilon > 0$ were arbitrary, the result follows from Riemann-Stieltjes Integral by Norm of Subdivision.
Category:Definite Integrals
Category:Riemann-Stieltjes Integral
Riemann-Stieltjes Integral of Constant Integrand

== Theorem ==
Let $\alpha$ be a real function that is bounded on $\closedint a b$.
Let $f : \closedint a b \to \R$ be defined as:
:$\map f x = c$
for some $c \in \R$.
Then, $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a b$, and:
:$\ds \int_a^b f \rd \alpha = c \map \alpha b - c \map \alpha a$
== Proof ==
Let $\epsilon > 0$
Define $P_\epsilon = \set {a, b}$.
Let $P = \set {x_0, \dotsc, x_n}$ be a subdivision of $\closedint a b$ that is finer than $P_\epsilon$.
Then:
Therefore:
Therefore, by definition of the Riemann-Stieltjes integral:
:$\ds \int_a^b f \rd \alpha = c \map \alpha b - c \map \alpha a$
== Sources ==
* : Chapter $7$ The Riemann-Stieltjes Integral: Exercise $7.1$
Category:Riemann-Stieltjes Integral
Riemann Hypothesis

== Hypothesis ==
All the nontrivial zeroes of the analytic continuation of the Riemann zeta function $\zeta$ have a real part equal to $\dfrac 1 2$.
=== Critical Strip ===
=== Critical Line ===
:
== Zeroes ==
== Also known as ==
== Also see ==
* All Nontrivial Zeroes of Riemann Zeta Function are on Critical Strip
* Critical Line Theorem: an infinite number of nontrivial zeroes exist on the critical line, whatever their multiplicity
== Historical Note ==
== Sources ==
* : $0,5$
* : $0 \cdotp 5$
* : Chapter $\text {A}.32$: Riemann ()
* : $0 \cdotp 5$
* : $14 \cdotp 13472 \, 5 \, \ldots$
* 
* 
* 
* 
* 
* 
* 
* : Appendix $18$: Millennium Prize problems: $3$.
* 
* : Appendix $23$: Millennium Prize problems: $3$.
Category:Riemann Hypothesis
Category:Unproven Hypotheses
Category:Riemann Zeta Function
Category:Millennium Problems
Category:Analytic Number Theory
Category:One Half