nyuuzyou/wikis huggingface dataset:
Laurent Expansion of Isolated Essential Singularity/Examples/Exponential of Reciprocal

== Examples of Use of Laurent Expansion of Isolated Essential Singularity ==
Let $f$ be the complex function defined as:
:$\forall z \in \C \setminus \set 0: \map f z = \map \exp {\dfrac 1 z}$
Then $f$ has an essential singularity at $z = 0$.
The Laurent expansion of $f$ is given by:
== Proof ==
Let $U \subseteq \C$ be an open set.
Let $f : U \to \C$ be a complex function.
So $f$ has a point at $z=0 $, for which $f$ is not analytic.
We have that $f$ has a singular point at $z = 0$.
$f$ cannot be extended to a holomorphic function $f: U \to \C$
So $f$ has a singular point at $z = 0$ which is not a removable singularity.
We have that $f$ has a singular point at $z = 0$ which is not a pole.
Therefore $f$ has a singular point which is neither a removable singularity nor a pole at $z = 0$
Hence the result by definition of removable singularity.
== Sources ==
* 
* 
Category:Laurent Expansion of Isolated Essential Singularity
Laurent Series Expansion for Cotangent Function

== Theorem == 
where:
:$z \in \C$ such that $\cmod z < 1$
:$\zeta$ is the Riemann Zeta function.
== Proof == 
From Mittag-Leffler Expansion for Cotangent Function:
Factoring $-\dfrac 1 {k^2}$: 
Taking $\cmod z < 1$, and noting that $k \ge 1$, we have, by Sum of Infinite Geometric Sequence:
:$\ds \pi \cot \pi z = \frac 1 z - 2 \sum_{k \mathop = 1}^\infty \frac z {k^2} \cdot \sum_{n \mathop = 1}^\infty \paren {\frac {z^2} {k^2} }^{n - 1}$
from which:
== Also see ==
* Power Series Expansion for Cotangent Function
Category:Laurent Series Expansion for Cotangent Function
Category:Laurent Series Expansions
Category:Cotangent Function
Category:Riemann Zeta Function at Even Integers
Law of Cosines

== Theorem ==
Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$c^2 = a^2 + b^2 - 2 a b \cos C$
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Also known as ==
== Also see ==
* Law of Sines
* Law of Tangents
* Spherical Law of Cosines
== Historical Note ==
== Sources ==
* : Chapter $\text I$: Properties of the Triangle
* : $\text V$. Trigonometry: Formulae $(7)$
* : $\S 5$: Trigonometric Functions: $5.93$
* 
* 
* 
* 
* 
* : Chapter $5$: Eternal Triangles: Logarithms
* : The cosine rule
* : The cosine rule
Category:Law of Cosines
Category:Triangles
Category:Cosine Function
Category:Named Theorems
Law of Cosines/Also known as

== Law of Cosines: Also known as ==
The Law of Cosines is also known as the Cosine Rule or Cosine Law.
It is known in France as Théorème d'Al-Kashi (Al-Kashi's Theorem) after , who is believed to have first discovered it.
== Sources ==
* 
* 
* 
* : The cosine rule
* : The cosine rule
Category:Law of Cosines
Law of Cosines/Proof 1

== Theorem ==
== Proof ==
Let $\triangle ABC$ be embedded in a Cartesian coordinate system by identifying:
:$C := \tuple {0, 0}$
:$B := \tuple {a, 0}$
:
Thus by definition of sine and cosine:
:$A = \tuple {b \cos C, b \sin C}$
By the Distance Formula:
:$c = \sqrt {\paren {b \cos C - a}^2 + \paren {b \sin C - 0}^2}$
Hence:
== Also known as ==
Category:Law of Cosines
Law of Cosines/Proof 2

== Theorem ==
== Proof ==
Let $\triangle ABC$ be a triangle.
=== Case 1: $AC$ greater than $AB$ ===
Using $AC$ as the radius, we construct a circle whose center is $A$.
Now we extend:
:$CB$ to $D$
:$AB$ to $F$
:$BA$ to $G$
:$CA$ to $E$.
$D$ is joined with $E$, thus:
:
Using the Intersecting Chords Theorem we have:
:$GB \cdot BF = CB \cdot BD$
$AF$ is a radius, so $AF = AC = b = GA$ and thus:
:$GB = GA + AB = b + c$
:$BF = AF - AB = b - c$
Thus:
Next:
As $CA$ is a radius, $CE$ is a diameter.
By Thales' Theorem, it follows that $\angle CDE$ is a right angle.
Then using the definition of cosine, we have
=== Case 2: $AC$ less than $AB$ ===
When $AC$ is less than $AB$, the point $B$ lies outside the circle and so the diagram needs to be modified accordingly:
:
Now we extend:
:$BA$ to $G$
:$CA$ to $E$.
Then we construct:
:$D$ as the point at which $CB$ intersects the circle
:$F$ as the point at which $AB$ intersects the circle.
Finally $D$ is joined to $E$.
Using the Secant Secant Theorem we have:
:$GB \cdot BF = CB \cdot BD$
$AF$ is a radius, so $AF = AC = b = GA$ and thus:
:$GB = GA + AB = b + c$
:$BF = AB - AF = b - c$
Thus:
Next:
As $CA$ is a radius, $CE$ is a diameter.
By Thales' Theorem, it follows that $\angle CDE$ is a right angle.
Then using the definition of cosine, we have
=== Case 3: $AC = AB$ ===
When $AC = AB$ the points $B$, $D$ and $F$ coincide on the circumference of the circle:
:
We extend:
:$BA$ to $G$
:$CA$ to $E$
and immediately:
:$GB = CB$
== Also known as ==
Category:Law of Cosines
Law of Cosines/Proof 3/Acute Triangle

== Theorem ==
Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that:
:$a$ is opposite $A$
:$b$ is opposite $B$
:$c$ is opposite $C$.
Let $\triangle ABC$ be an acute triangle.
Then:
:$c^2 = a^2 + b^2 - 2a b \cos C$
== Proof ==
Let $\triangle ABC$ be an acute triangle.
:
Let $BD$ be dropped perpendicular to $AC$.
Let:
We have that $\triangle CDB$ and $\triangle ADB$ are right triangles.
Hence:
Then:
== Sources ==
* : $\text V$. Trigonometry: The cos formula and the sine formula
Category:Law of Cosines
Law of Cosines/Proof 3/Obtuse Triangle

== Theorem ==
Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that:
: $a$ is opposite $A$
: $b$ is opposite $B$
: $c$ is opposite $C$.
Let $\triangle ABC$ be an obtuse triangle such that $A$ is obtuse
Then:
:$c^2 = a^2 + b^2 - 2a b \cos C$
== Proof ==
Let $\triangle ABC$ be an obtuse triangle.
:
Let $AC$ be extended and $BD$ be dropped perpendicular to $AC$.
Let:
We have that $\triangle CDB$ and $\triangle ADB$ are right triangles.
Hence:
Then:
== Sources ==
* : $\text V$. Trigonometry: The cos formula and the sine formula
Category:Law of Cosines
Law of Cosines/Right Triangle

== Theorem ==
Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that:
: $a$ is opposite $A$
: $b$ is opposite $B$
: $c$ is opposite $C$.
Let $\triangle ABC$ be a right triangle such that $\angle A$ is right.
Then:
:$c^2 = a^2 + b^2 - 2 a b \cos C$
== Proof ==
Let $\triangle ABC$ be a right triangle such that $\angle A$ is right.
:
Category:Law of Cosines
Law of Division

== Theorem ==
Let $\mathbb F$ denote one of the following number systems:
: rational numbers $\Q$
: real numbers $\R$
: complex numbers $\C$
Let $a, b \in \mathbb F$ such that $a \ne 0$.
Then there exists a unique $x$ such that:
:$a x = b$
$x$ is then defined and denoted:
:$x := b / a$
== Proof ==
== Sources ==
* : $\S 1.1$. Number Systems: $\text{VI}.$
Category:Numbers
Law of Excluded Middle

== Proof Rule ==
The law of (the) excluded middle is a valid argument in certain types of logic dealing with disjunction $\lor$ and negation $\neg$.
This includes classical propositional logic and predicate logic, and in particular natural deduction, but for example not intuitionistic propositional logic.
=== Proof Rule ===
=== Sequent Form ===
== Explanation ==
== Intuitionist Perspective ==
== Also known as ==
== Also see ==
* Definition:Disjunction
* Definition:Logical Not
* Principle of Non-Contradiction
== Sources ==
* : Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic
* : $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Example $1.6 \ \text{(a)}$
* 
* : Chapter $1$: Introduction: $\S 1.4$: Non-standard logics
* : $2$. Graphs: Paradox
* 
* 
* 
* 
* 
* 
* 
Category:Law of Excluded Middle
Category:Disjunction
Category:Tautology
Category:Rules of Inference
Category:Philosophical Positions
Law of Excluded Middle/Also known as

== Law of Excluded Middle: Also known as ==
The law of (the) excluded middle is otherwise known as:
* (Principium) tertium non datur, Latin for third not given, that is, a third possibility is not possible
* Principium tertii exclusi, Latin for the Principle of the Excluded Third (PET).
* the principle of bivalence.
== Sources ==
* 
* 
* 
* 
Category:Law of Excluded Middle
Law of Excluded Middle/Explanation

== Proof Rule ==
The law of (the) excluded middle can be expressed in natural language as:
:Every statement is either true or false.
This is one of the Aristotelian principles upon which rests the whole of classical logic, and the majority of mainstream mathematics.
== Sources ==
* 
* : $\S 1$: Informal statement calculus: $\S 1.1$: Statements and connectives (in passing)
Category:Law of Excluded Middle
Law of Excluded Middle/Intuitionist Perspective

== Law of Excluded Middle: Intuitionist Perspective == 
The Law of the Excluded Middle is rejected by the intuitionistic school, which rejects the existence of an object unless it can be constructed within an axiomatic framework which does not include the Law of the Excluded Middle.
This is because from the perspective of intuitionism:
:an object cannot be known to exist unless it can be constructed in a finite number of steps
:a statement cannot be known to be true if its proof needs an argument requiring an infinite number of steps.
Thus:
:while it is sufficient to prove a statement is not true by demonstrating that it is not true
:it is insufficient to prove a statement is true by demonstrating it is not false.
== Sources ==
* 
* 
Category:Law of Excluded Middle
Law of Excluded Middle/Proof Rule

== Proof Rule ==
The law of (the) excluded middle is a valid argument in certain types of logic dealing with disjunction $\lor$ and negation $\neg$.
This includes classical propositional logic and predicate logic, and in particular natural deduction, but for example not intuitionistic propositional logic.
As a proof rule it is expressed in the form:
:$\phi \lor \neg \phi$ for all statements $\phi$.
It can be written:
:$\ds {{} \over \phi \lor \neg \phi} \textrm{LEM} \qquad \text { or } \qquad {\top \over \phi \lor \neg \phi} \textrm{LEM}$
where the symbol $\top$ (top) signifies tautology.
=== Tableau Form ===
== Explanation ==
== Also known as ==
== Also see ==
* This is a rule of inference of the following proof systems:
** Definition:Natural Deduction
== Technical Note ==
== Sources ==
* : $\S 1.2.2$: Derived rules
Category:Rules of Inference
Category:Law of Excluded Middle
Law of Excluded Middle/Proof Rule/Technical Note

== Technical Note on ==
When invoking in a tableau proof, use the template:
:{{ExcludedMiddle|line|statement}}
or:
:{{ExcludedMiddle|line|statement|comment}}
where:
:line is the number of the line on the tableau proof where the is to be invoked
:statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters
:comment is the (optional) comment that is to be displayed in the Notes column.
Category:Technical Notes
Law of Excluded Middle/Sequent Form

== Theorem ==
The Law of Excluded Middle can be symbolised by the sequent:
:$\vdash p \lor \neg p$
== Proof 1 ==
== Proof 2 ==
== Proof by Truth Table ==
== Sources ==
* : $\S \text{II}.13$: Symbolism of sentential calculus
* : $\S 3.6$: Reference Formulae: $RF \, 2$
* : $\S 4.7$: The Derivation of Formulae: $D \, 3$
* : $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$: Theorem $\text{T59}$
* : $\S 2.3.3$
Category:Law of Excluded Middle
Law of Excluded Middle/Sequent Form/Proof by Truth Table

== Theorem ==
== Proof ==
We apply the Method of Truth Tables to the proposition $\vdash p \lor \neg p$.
As can be seen by inspection, the truth value of the main connective, that is $\lor$, is $\T$ for each boolean interpretation for $p$.
:$\begin{array}{|c|c|cc|} \hline
p & \lor & \neg & p \\
\hline
\F & \T & \T & \F \\
\T & \T & \F & \T \\
\hline
\end{array}$
== Sources ==
* : $2$ Arguments Containing Compound Statements: $2.4$: Statement Forms
Category:Law of Excluded Middle
Law of Identity

== Theorem ==
Every proposition entails itself:
=== Formulation 1 ===
=== Formulation 2 ===
A seemingly trivial rule, but can be surprisingly useful to get a particular formula into the right place in a proof.
=== Interpretation by Models ===
Clearly, every model of $P$ is a model of $P$.
Thus by definition of semantic consequence:
: $P \models P$
== Also known as ==
This is also known as the rule of repetition.
== Also see ==
* Rule of Assumption
* Biconditional is Reflexive, where $\vdash p \iff p$ is shown.
== Technical Note ==
== Sources ==
* : $\S \text{II}.12$: Laws of sentential calculus
Category:Law of Identity
Category:Conditional
Category:Propositional Logic
Law of Identity/Formulation 1/Proof 1

== Theorem ==
== Proof ==
This is the shortest tableau proof possible.
== Sources ==
* : Chapter $1$: The Propositional Calculus $1$: $5$ Further Proofs: Résumé of Rules: Theorem $29$
* : $\S 1.2.1$: Rules for natural deduction
Category:Law of Identity
Law of Identity/Formulation 1/Proof by Truth Table

== Theorem ==
== Proof ==
We apply the Method of Truth Tables (trivially) to the proposition.
$\begin{array}{|c|c|} \hline
p & p \\
\hline
\F & \F \\
\T & \T \\
\hline
\end{array}$
Category:Law of Identity
Category:Truth Table Proofs
Law of Identity/Formulation 2

== Theorem ==
Every proposition entails itself:
:$\vdash p \implies p$
== Proof 1 ==
== Proof 2 ==
== Proof by Truth Table ==
== Also see ==
Some sources, for example , use the statement:
:$\vdash p \implies p$
to be the defining property of a tautology.
== Sources ==
* : Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations
* : $\S 3.6$: Reference Formulae: $RF \, 3$
* : $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
* : $\S \text{I}: 3$: Logical Constants $(2)$
* : $\S 1$: Informal statement calculus: $\S 1.3$: Rules for manipulation and substitution
Category:Law of Identity
Law of Identity/Formulation 2/Proof 1

== Theorem ==
== Proof ==
This is the second shortest tableau proof possible.
== Sources ==
* : $\text{I}$: 'NOT' and 'IF': $\S 5$: Theorem $\text{T1}$
* : Chapter $2$: The Propositional Calculus $2$: $2$: Theorems and Derived Rules: Theorem $38$
* : $\S 1.2.1$: Rules for natural deduction
Category:Law of Identity
Law of Identity/Formulation 2/Proof by Truth Table

== Theorem ==
== Proof ==
We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, the truth value under the main connective is $\T$ throughout.
$\begin{array}{|ccc|} \hline
p & \implies & p \\
\hline
\F & \T & \F \\
\T & \T & \T \\
\hline
\end{array}$
== Sources ==
* : Chapter $2$: The Propositional Calculus $2$: $3$ Truth-Tables: Exercise $1 \ \text{(i) (a)}$
Category:Law of Identity
Category:Truth Table Proofs
Law of Identity/Technical Note

== Technical Note on Law of Identity ==
When invoking the Law of Identity in a tableau proof, use the template:
:{{IdentityLaw|line|pool|depends|statement}}
or:
:{{IdentityLaw|line|pool|depends|statement|comment}}
where:
:line is the number of the line on the tableau proof where Law of Identity is to be invoked
:pool is the pool of assumptions (comma-separated list)
:statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters
:depends is the line of the tableau proof upon which this line directly depends
:comment is the (optional) comment that is to be displayed in the Notes column.
Category:Technical Notes
Law of Inverses (Modulo Arithmetic)

== Theorem ==
Let $m, n \in \Z$.
Then:
:$\exists n' \in \Z: n n' \equiv d \pmod m$
where $d = \gcd \set {m, n}$.
=== Corollary 1 ===
=== Corollary 2 ===
== Proof ==
We have that $d = \gcd \set {m, n}$.
So:
So $b$ (in the above) fits the requirement for $n'$ in the assertion to be proved.
Category:Modulo Arithmetic
Category:Named Theorems
Law of Inverses (Modulo Arithmetic)/Corollary 1

== Corollary to Law of Inverses (Modulo Arithmetic) ==
Let $m, n \in \Z$ such that:
:$m \perp n$
that is, such that $m$ and $n$ are coprime.
Then:
:$\exists n' \in \Z: n n' \equiv 1 \pmod m$
== Proof ==
By definition of coprime:
:$m \perp n \iff \gcd \set {m, n} = 1$
The result follows directly from Law of Inverses (Modulo Arithmetic).
== Sources ==
* : $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $19$
Category:Modulo Arithmetic
Law of Inverses (Modulo Arithmetic)/Corollary 2

== Corollary to Law of Inverses (Modulo Arithmetic) ==
Let $m, n \in \Z$ such that:
:$m \perp n$
that is, such that $m$ and $n$ are coprime.
Then:
:$\exists n' \in \Z: n n' \equiv 1 \pmod m$
where:
:$n' \equiv n^{\map \phi n - 1} \pmod m$
where $\map \phi n$ is the Euler $\phi$ function.
== Proof ==
From :
:$\exists n' \in \Z: n n' \equiv 1 \pmod m$
From Euler's Theorem (Number Theory):
:$n^{\map \phi m} \equiv 1 \pmod m$
it follows that:
:$n \cdot n^{\map \phi m - 1} \equiv 1 \pmod m$
The result follows directly.
== Sources ==
* : $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $28$
Category:Modulo Arithmetic
Law of Mass Action

== Theorem ==
Let $\AA$ and $\BB$ be two chemical substances in a solution $C$ which are involved in a second-order reaction.
Let $x$ grams of $\CC$ contain $a x$ grams of $\AA$ and $b x$ grams of $\BB$, where $a + b = 1$.
Let there be $a A$ grams of $\AA$ and $b B$ grams of $\BB$ at time $t = t_0$, at which time $x = 0$.
Then:
:$x = \begin{cases}
\dfrac {k A^2 a b t} {k A a b t + 1} & : A = B \\
& \\
\dfrac {A B e^{-k \paren {A - B} a b t}} {A - B e^{-k \paren {A - B} a b t}} & : A \ne B
\end{cases}$
for some positive real constant $k$.
This is known as the law of mass action.
== Proof ==
By the definition of a second-order reaction:
:The rate of formation of $\CC$ is jointly proportional to the quantities of $\AA$ and $\BB$ which have not yet transformed.
By definition of joint proportion:
:$\dfrac {\d x} {\d t} \propto \paren {A - x} a \paren {B - x} b$
or:
:$\dfrac {\d x} {\d t} = k a b \paren {A - x} \paren {B - x}$
for some positive real constant $k$.
Thus:
Note that in the above, we have to assume that $A \ne B$ or the integrals on the will not be defined.
We will look later at how we handle the situation when $A = B$.
We are given the initial condition $x = 0$ at $t = 0$.
Thus:
Our assumption that $C \ne 1$ is justified, because that only happens when $A = B$, and we have established that this is not the case.
So, we now have:
Now we can investigate what happens when $A = B$.
We need to solve:
:$\dfrac {\d x} {\d t} = k a b \paren {A - x} \paren {A - x} = k a b \paren {A - x}^2$
So:
We are given the initial condition $x = 0$ at $t = 0$.
Thus:
This gives us:
So:
:$x = \begin{cases}
\dfrac {k A^2 a b t} {k A a b t + 1} & : A = B \\
& \\
\dfrac {A B e^{-k \paren {A - B} a b t} } {A - B e^{-k \paren {A - B} a b t} } & : A \ne B
\end{cases}$
== Sources ==
* : $1$: The Nature of Differential Equations: $\S 4$: Growth, Decay and Chemical Reactions: Problem $1$
Category:Chemistry
Category:First Order ODEs
Category:Named Theorems
Law of Quadratic Reciprocity

== Theorem ==
Let $p$ and $q$ be distinct odd primes.
Then:
:$\paren {\dfrac p q} \paren {\dfrac q p} = \paren {-1}^{\dfrac {\paren {p - 1} \paren {q - 1} } 4}$
where $\paren {\dfrac p q}$ and $\paren {\dfrac q p}$ are defined as the Legendre symbol.
An alternative formulation is:
$\paren {\dfrac p q} = \begin{cases}
\quad \paren {\dfrac q p} & : p \equiv 1 \lor q \equiv 1 \pmod 4 \\
-\paren {\dfrac q p} & : p \equiv q \equiv 3 \pmod 4
\end{cases}$
The fact that these formulations are equivalent is immediate.
This fact is known as the Law of Quadratic Reciprocity, or LQR for short.
=== First Supplement to Law of Quadratic Reciprocity ===
=== Second Supplement to Law of Quadratic Reciprocity ===
== Proof ==
Let $p$ and $q$ be distinct odd primes.
Consider the rectangle in the Cartesian $x y$ plane with vertices at $\tuple {0, 0}, \tuple {\dfrac p 2, 0}, \tuple {\dfrac p 2, \dfrac q 2}, \tuple {0, \dfrac q 2}$.
The number of lattice points inside this rectangle is $\dfrac {p - 1} 2 \times \dfrac {q - 1} 2$.
:
Consider the diagonal from $\tuple {0, 0}$ to $\tuple {\dfrac p 2, \dfrac q 2}$.
This has the equation:
:$y = \dfrac q p x$
 there were a lattice point $\tuple {a, b}$ on the diagonal.
Then:
:$b = \dfrac q p a$
But as $p$ and $q$ are distinct primes, this would mean:
:$p$ divides $a$
and:
:$q$ divides $b$
This means that $\tuple {a, b}$ has to be outside the rectangle.
So there are no lattice points on the diagonal inside the rectangle.
Let $A$ and $B$ be the triangular regions inside the rectangle lying respectively above and below the diagonal.
Let $k \in \Z: 0 < k < \dfrac p 2$.
The number of lattice points in $B$ which lie directly above the point $\tuple {k, 0}$ is:
:$\floor {\dfrac q p k}$
where $\floor {\dfrac q p k}$ is the floor of $\dfrac q p k$.
So the total number of lattice points in $B$ is given by:
:$\ds N_B = \sum_{k \mathop = 1}^{\frac{p - 1} 2} \floor {\dfrac q p k}$
Let $\map \alpha {q, p}$ be defined as:
:$\ds \map \alpha {q, p} = \sum_{k \mathop = 1}^{\frac {p - 1} 2} \floor {\dfrac q p k}$
In the same way, by counting the lattice points to the right of $\tuple {0, k}$, the total number of lattice points in $A$ is given by:
:$N_A = \map \alpha {p, q}$
Now the total number of lattice points in the rectangle is $N_A + N_B$.
But this is also equal to:
:$\dfrac {p - 1} 2 \times \dfrac {q - 1} 2 = \dfrac {\paren {p - 1} \paren {q - 1} } 4$
So we have that:
:$\map \alpha {p, q} + \map \alpha {q, p} = \dfrac {\paren {p - 1} \paren {q - 1} } 4$
The fact that these counts are equal relies upon the fact that there are no lattice points on the diagonal, as demonstrated above.
Now we invoke Eisenstein's Lemma:
:$\paren {\dfrac a p} = \paren {-1}^{\map \alpha {a, p} }$
So this gives us:
 
which is LQR.
== Also known as ==
== Historical Note ==
== Sources ==
* 
* 
* : $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $47 \ \text{d)}$
* 
* 
* : Chapter $7$: Patterns in Numbers: Gauss
Category:Law of Quadratic Reciprocity
Category:Legendre Symbol
Category:Named Theorems
Category:Quadratic Residues
Law of Quadratic Reciprocity/Historical Note

== Historical Note on Law of Quadratic Reciprocity ==
The Law of Quadratic Reciprocity was investigated by who stated it imperfectly and failed to find a proof.
 first stated it correctly, but his attempted proof of it was incorrect.
 first became aware of this result, and found a correct proof. He wrote this up in and published it in his in .
The proof as given here was found by and published in .
== Sources ==
* : $\S 3$: Appendix $\text A$: Euler
* : Chapter $\text {A}.25$: Gauss ()
* : Chapter $7$: Patterns in Numbers: Gauss
Category:Historical Notes
Law of Simple Conversion of E

== Theorem ==
Consider the universal negative categorical statement No $S$ is $P$:
:$\map {\mathbf E} {S, P}: \forall x: \map S x \implies \neg \map P x$
Then No $P$ is $S$:
:$\map {\mathbf E} {P, S}$
== Proof ==
== Sources ==
* : Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism
Category:Laws of Conversion
Category:Universal Negative
Category:Categorical Statements
Law of Simple Conversion of I

== Theorem ==
Consider the particular affirmative categorical statement Some $S$ is $P$:
:$\map {\mathbf I} {S, P}: \exists x: \map S x \land \map P x$
Then Some $P$ is $S$:
:$\map {\mathbf I} {P, S}$
== Proof ==
== Sources ==
* : Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism
Category:Laws of Conversion
Category:Particular Affirmative
Category:Categorical Statements
Law of Sines

== Theorem ==
Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$
where $R$ is the circumradius of $\triangle ABC$.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Also presented as ==
== Also known as ==
== Also see ==
* Law of Cosines
* Law of Tangents
* Spherical Law of Sines
== Historical Note ==
== Sources ==
* : $\text V$. Trigonometry: Formulae $(8)$
* : $\S 5$: Trigonometric Functions: $5.92$
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* : Chapter $5$: Eternal Triangles: Logarithms
* : The sine rule
* : The sine rule
Category:Law of Sines
Category:Triangles
Category:Sine Function
Category:Named Theorems