nyuuzyou/wikis huggingface dataset:
Laplace Transform of Error Function of Root

== Theorem ==
:$\laptrans {\map \erf {\sqrt t} } = \dfrac 1 {s \sqrt {s + 1} }$
where:
:$\laptrans f$ denotes the Laplace transform of the function $f$
:$\erf$ denotes the error function
== Proof 1 ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Laplace Transforms of Special Functions: $7$
Category:Laplace Transforms
Category:Error Function
Category:Laplace Transform of Error Function of Root
Laplace Transform of Exponential/Examples/2 e^4 t

== Example of Use of Laplace Transform of Exponential ==
:$\laptrans {2 e^{4 t} } = \dfrac 2 {s - 4}$
for $s > 4$.
== Proof ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Supplementary Problems: Laplace Transforms of Elementary Functions: $51 \ \text {(a)}$
Category:Examples of Laplace Transforms
Laplace Transform of Exponential/Real Argument

== Theorem ==
Let $\laptrans f$ denote the Laplace transform of a function $f$.
Let $e^x$ be the real exponential.
Then:
:$\map {\laptrans {e^{a t} } } s = \dfrac 1 {s - a}$
where $a \in \R$ is constant, and $\map \Re s > \map \Re a$.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Laplace Transforms of some Elementary Functions: $4$.
* : $\S 32$: Table of Special Laplace Transforms: $32.29$
Category:Laplace Transforms‎
Category:Exponential Function
Category:Laplace Transform of Exponential
Laplace Transform of Exponential/Real Argument/Proof 1

== Theorem ==
== Proof ==
Because $s > a$, we have that $a - s < 0$.
Hence:
:$\ds \lim_{L \mathop \to \infty} \paren {a - s} L \to -\infty$
So:
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Laplace Transforms of some Elementary Functions: $1 \ \text{(c)}$
Category:Laplace Transform of Exponential
Laplace Transform of Exponential/Real Argument/Proof 3

== Theorem ==
== Proof ==
From Laplace Transform of Derivative:
:$(1): \quad \laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f 0$
under suitable conditions.
Then:
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Laplace Transform of Derivative: $15 \ \text{(c)}$
Category:Laplace Transform of Exponential
Laplace Transform of Exponential Integral Function

== Theorem ==
Let $\Ei: \R_{>0} \to \R$ denote the exponential integral function:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
:$\laptrans {\map \Ei t} = \dfrac {\map \ln {s + 1} } s$
where $\laptrans f$ denotes the Laplace transform of the function $f$
== Proof 1 ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Laplace Transforms of Special Functions: $10$
Category:Laplace Transforms
Category:Exponential Integral Function
Category:Laplace Transform of Exponential Integral Function
Laplace Transform of Exponential Integral Function/Proof 1

== Theorem ==
== Proof ==
Let $\map f t := \map \Ei t = \ds \int_t^\infty \dfrac {e^{-u} } u \rd u$.
Then:
By the Initial Value Theorem of Laplace Transform:
:$\ds \lim_{s \mathop \to \infty} s \laptrans {\map f t} = \lim_{t \mathop \to 0} \map f t = \map f 0 = 0$
which leads to:
:$c = 0$
Thus:
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: The Sine, Cosine and Exponential Integrals: $38$
Category:Laplace Transform of Exponential Integral Function
Laplace Transform of Exponential times Cosine

== Theorem ==
:$\map {\laptrans {e^{b t} \cos a t} } s = \dfrac {s - b} {\paren {s - b}^2 + a^2}$
where:
:$a$ and $b$ are real numbers
:$s$ is a complex number with $\map \Re s > a + b$
:$\laptrans f$ denotes the Laplace transform of $f$ evaluated at $s$.
== Proof ==
== Sources ==
* : $\S 32$: Table of Special Laplace Transforms: $32.35$
Category:Laplace Transforms‎
Category:Exponential Function
Category:Cosine Function
Laplace Transform of Exponential times Function

== Theorem ==
Let $\map f t: \R \to \R$ or $\R \to \C$ be a function of exponential order $a$ for some constant $a \in \R$.
Let $\laptrans {\map f t} = \map F s$ be the Laplace transform of $f$.
Let $e^t$ be the exponential function.
Then:
:$\laptrans {e^{a t} \map f t} = \map F {s - a}$
everywhere that $\laptrans f$ exists, for $\map \Re s > a$
== Proof ==
== Also known as ==
This property of the Laplace transform operator is sometimes seen referred to as:
:the first translation property
or:
:the first shifting property.
== Examples ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $2$. First translation or shifting property: Theorem $1 \text{-} 3$
* : Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $7$
* : $\S 32$: Table of General Properties of Laplace Transforms: $32.5$
* 
Category:Laplace Transforms
Category:Exponential Function
Laplace Transform of Exponential times Function/Examples

== Examples of Use of Laplace Transform of Exponential times Function ==
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
=== Example $1$ ===
=== Example $2$ ===
=== Example $3$ ===
=== Example $4$ ===
=== Example $5$ ===
Category:Examples of Laplace Transforms
Laplace Transform of Exponential times Function/Examples/Example 1

== Examples of Use of Laplace Transform of Exponential times Function ==
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
:$\laptrans {e^{-t} \cos 2 t} = \dfrac {s + 1} {s^2 + 2 s + 5}$
== Proof ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $2$. First translation or shifting property
Category:Examples of Laplace Transforms
Laplace Transform of Exponential times Function/Examples/Example 2

== Examples of Use of Laplace Transform of Exponential times Function ==
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
:$\laptrans {t^2 e^{3 t} } = \dfrac 2 {\paren {s - 3}^3}$
== Proof ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $8 \ \text{(a)}$
Category:Examples of Laplace Transforms
Laplace Transform of Exponential times Function/Examples/Example 3

== Examples of Use of Laplace Transform of Exponential times Function ==
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
:$\laptrans {e^{-2 t} \sin 4 t} = \dfrac 4 {s^2 + 4 s + 20}$
== Proof ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $8 \ \text{(b)}$
Category:Examples of Laplace Transforms
Laplace Transform of Exponential times Function/Examples/Example 4

== Example of Use of Laplace Transform of Exponential times Function ==
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
:$\laptrans {e^{4 t} \cosh 5 t} = \dfrac {s - 4} {s^2 - 8 s - 9}$
== Proof 1 ==
== Proof 2 ==
Category:Examples of Laplace Transforms
Category:Laplace Transform of Exponential times Function/Examples/Example 4
Laplace Transform of Exponential times Function/Examples/Example 4/Proof 1

== Example of Use of Laplace Transform of Exponential times Function ==
== Proof ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $8 \ \text{(c)}$
Category:Laplace Transform of Exponential times Function/Examples/Example 4
Laplace Transform of Exponential times Function/Examples/Example 4/Proof 2

== Example of Use of Laplace Transform of Exponential times Function ==
== Proof ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $8 \ \text{(c)}$
Category:Laplace Transform of Exponential times Function/Examples/Example 4
Laplace Transform of Exponential times Function/Examples/Example 5

== Examples of Use of Laplace Transform of Exponential times Function ==
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
:$\laptrans {e^{-3 t} \paren {3 \cos 6 t - 5 \sin 6 t} } = \dfrac {3 s - 24} {s^2 + 4 s + 40}$
== Proof ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $8 \ \text{(d)}$
Category:Examples of Laplace Transforms
Laplace Transform of Exponential times Sine

== Theorem ==
:$\map {\laptrans {e^{b t} \sin a t} } s = \dfrac a {\paren {s - b}^2 + a^2}$
where:
:$a$ and $b$ are real numbers
:$s$ is a complex number with $\map \Re s > a + b$
:$\laptrans f$ denotes the Laplace transform of $f$ evaluated at $s$.
== Proof ==
== Sources ==
* : $\S 32$: Table of Special Laplace Transforms: $32.34$
Category:Laplace Transforms‎
Category:Exponential Function
Category:Sine Function
Laplace Transform of Function of t minus a

== Theorem ==
Let $f$ be a function such that $\laptrans f$ exists.
Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$.
Let $a \in \C$ or $\R$ be constant.
Let $g$ be the function defined as:
:$\map g t = \begin{cases} \map f {t - a} & : t > a \\ 0 & : t \le a \end{cases}$
Then:
:$\laptrans {\map g t} = e^{-a s} \map F s$
== Proof 1 ==
== Proof 2 ==
== Also known as ==
This property of the Laplace transform operator is sometimes seen referred to as:
:the second translation property
or:
:the second shifting property.
== Examples ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $3$. Second translation or shifting property: Theorem $1 \text{-} 4$
* : $\S 32$: Table of General Properties of Laplace Transforms: $32.6$
Category:Laplace Transforms
Category:Laplace Transform of Function of t minus a
Laplace Transform of Function of t minus a/Examples/Example 1

== Example of Use of Laplace Transform of Function of t minus a ==
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
:$\laptrans {\paren {t - 2}^3} = \dfrac {6 e^{-2 s} } {s^4}$
where $t > 2$.
== Proof ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $3$. Second translation or shifting property
Category:Examples of Laplace Transforms
Laplace Transform of Function of t minus a/Examples/Example 2

== Example of Use of Laplace Transform of Function of t minus a ==
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Let $f: \R \to \R$ be the function defined as:
:$\forall t \in \R: \map f t = \begin {cases} \map \cos {t - \dfrac {2 \pi} 3} & : t \ge \dfrac {2 \pi} 3 \\ 0 & : t < \dfrac {2 \pi} 3 \end {cases}$
Then:
:$\laptrans {\map f t} = s \exp \dfrac {-2 \pi s} 3 \dfrac 1 {s^2 + 1}$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $10$
Category:Examples of Laplace Transforms
Category:Laplace Transform of Function of t minus a/Examples/Example 2
Laplace Transform of Function of t minus a/Examples/Example 2/Proof 1

== Example of Use of Laplace Transform of Function of t minus a ==
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
== Proof ==
and the result follows.
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $10$
Category:Laplace Transform of Function of t minus a/Examples/Example 2
Laplace Transform of Function of t minus a/Examples/Example 2/Proof 2

== Example of Use of Laplace Transform of Function of t minus a ==
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
== Proof ==
and the result follows.
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $10$
Category:Laplace Transform of Function of t minus a/Examples/Example 2
Laplace Transform of Function of t minus a/Proof 1

== Theorem ==
Let $f$ be a function such that $\laptrans f$ exists.
Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$.
Let $a \in \C$ or $\R$ be constant.
== Proof ==
Category:Laplace Transform of Function of t minus a
Laplace Transform of Function of t minus a/Proof 2

== Theorem ==
Let $f$ be a function such that $\laptrans f$ exists.
Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$.
Let $a \in \C$ or $\R$ be constant.
== Proof ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $9$
Category:Laplace Transform of Function of t minus a
Laplace Transform of Generating Function of Sequence

== Theorem ==
Let $\sequence{a_n}$ be a sequence which has a generating function which is convergent.
Let $\map G z$ be the generating function for $\sequence{a_n}$.
Let $\map f x$ be the step function:
:$\map f x = \ds \sum_{k \mathop \in \Z} a_k \sqbrk{0 \le k \le x}$
where $\sqbrk{0 \le k \le x}$ is Iverson's convention.
Then the Laplace transform of $\map f x$ is given by:
:$\laptrans {\map f s} = \dfrac {\map G {e^{-s} }} s$
== Proof ==
We note that:
and:
Then:
== Sources ==
* : $\S 1.2.9$: Generating Functions: Exercise $13$
Category:Generating Functions
Category:Laplace Transforms
Laplace Transform of Half Wave Rectified Sine Curve

== Theorem ==
Consider the half wave rectified sine curve:
:$\map f t = \begin {cases} \sin t & : 2 n \pi \le t \le \paren {2 n + 1} \pi \\ 0 & : \paren {2 n + 1} \pi \le t \le \paren {2 n + 2} \pi \end {cases}$
The Laplace transform of $\map f t$ is given by:
:$\laptrans {\map f t} = \dfrac 1 {\paren {1 - e^{-\pi s} } \paren {s^2 + 1} }$
== Proof ==
We have that $\map f t$ is periodic with period $2 \pi$:
:
Hence:
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Periodic Functions: $24 \ \text{(b)}$
Category:Laplace Transforms
Category:Half Wave Rectified Sine Curve
Laplace Transform of Heaviside Step Function

== Theorem ==
Let $\map {u_c} t$ denote the Heaviside step function:
:$\map {u_c} t = \begin{cases}
1 & : t > c \\
0 & : t < c
\end{cases}$
The Laplace transform of $\map {u_c} t$ is given by:
:$\laptrans {\map {u_c} t} = \dfrac {e^{-s c} } s$
for $\map \Re s > c$.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Laplace Transforms of Special Functions: $11$
Category:Laplace Transforms
Category:Heaviside Step Function‎
Category:Laplace Transform of Heaviside Step Function
Laplace Transform of Heaviside Step Function times Function

== Theorem ==
Let $\map f t: \R \to \R$ or $\R \to \C$ be a function of exponential order $a$ for some constant $a \in \R$.
Let $f$ be piecewise continuous with one-sided limits on any closed interval of the form $\closedint 0 b$ where $b > 0$.
Let $\map {u_c} t$ be the Heaviside step function.
Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$.
Then:
:$\laptrans {\map {u_c} t \, \map f {t - c} } = e^{-s c} \map F s$
for $\map \Re s > a$.
== Proof ==
Let $u = t - c$.
Then $\dfrac {\rd u} {\rd t} = 1$.
Then $u \to 0^-$ as $t \to c^+$.
Also, $u \to +\infty$ as $t \to +\infty$.
So:
== Sources ==
* 
Category:Laplace Transforms
Category:Heaviside Step Function‎
Laplace Transform of Higher Order Derivatives

== Theorem ==
Let $f: \R \to \R$ or $\R \to \C$ be a continuous function on any interval of the form $0 \le t \le a$.
Let $f$ be $n$ times differentiable on said intervals.
Let $f, f', \ldots, f^{\paren {n - 1} }$ be continuous and $f^{\paren n}$ piecewise continuous with one-sided limits on said intervals.
Let $f, f', \ldots, f^{\paren {n - 1} }$ be of exponential order.
Let $\laptrans f = F$ denote the Laplace transform of $f$.
Then $\laptrans {f^{\paren n} }$ exists for $\map \Re s > a$, and:
== Proof ==
The proof proceeds by induction on $n$, the order of the derivative of $f$.
=== Basis for the Induction ===
By hypothesis $f$ is of exponential order.
Thus Laplace Transform of Derivative can be invoked:
:$\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f 0$
This is the basis for the induction.
=== Induction Hypothesis ===
Fix $n - 1 \in \N$ with $n \ge 2$.
Assume:
:$\ds \laptrans {\map {f^{\paren {n - 1} } } t} = s^{n - 1} \laptrans {\map f t}- \sum_{j \mathop = 1}^{n - 1} s^{j - 1} \map {f^{\paren {n - 1 - j} } } 0$
This is our induction hypothesis.
=== Induction Step ===
This is our induction step:
By hypothesis $f^\paren {n - 1}$ is of exponential order.
Thus Laplace Transform of Derivative can be invoked:
The result follows by the Principle of Mathematical Induction.
== Also see ==
* Higher Order Derivatives of Laplace Transform
== Sources ==
* : Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $5$. Laplace transform of derivatives: Theorem $1 \text{-} 10$
* : $\S 32$: Table of General Properties of Laplace Transforms: $32.9$
* : $\S 6.2$
Category:Laplace Transforms of Derivatives
Laplace Transform of Hyperbolic Cosine

== Theorem ==
Let $\cosh t$ be the hyperbolic cosine, where $t$ is real.
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Then:
:$\laptrans {\cosh a t} = \dfrac s {s^2 - a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > a$.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Laplace Transforms of some Elementary Functions: $8$.
* : $\S 32$: Table of Special Laplace Transforms: $32.37$
Category:Hyperbolic Cosine Function
Category:Laplace Transforms
Category:Laplace Transform of Hyperbolic Cosine
Laplace Transform of Hyperbolic Sine

== Theorem ==
Let $\sinh t$ be the hyperbolic sine, where $t$ is real.
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Then:
:$\laptrans {\sinh a t} = \dfrac a {s^2 - a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > a$.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Laplace Transforms of some Elementary Functions: $7$.
* : $\S 32$: Table of Special Laplace Transforms: $32.36$
Category:Hyperbolic Sine Function
Category:Laplace Transforms
Category:Laplace Transform of Hyperbolic Sine
Laplace Transform of Identity Mapping

== Theorem ==
Let $\laptrans f$ denote the Laplace transform of a function $f$.
Let $\map {I_\R} t$ denote the identity mapping on $\R$ for $t > 0$.
Then:
:$\laptrans {\map {I_\R} t} = \dfrac 1 {s^2}$
for $\map \Re s > 0$.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Laplace Transforms of some Elementary Functions: $2$.
* : $\S 32$: Table of Special Laplace Transforms: $32.26$
Category:Laplace Transforms
Category:Identity Mappings
Category:Laplace Transform of Identity Mapping
Laplace Transform of Identity Mapping/Proof 2

== Theorem ==
== Proof ==
From Integration by Parts:
:$\ds \int f g' \rd t = f g - \int f'g \rd t$
Here:
So: 
Evaluating at $t = 0$ and $t \to +\infty$:
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Laplace Transforms of some Elementary Functions: $1 \ \text{(b)}$
* 
Category:Laplace Transform of Identity Mapping
Laplace Transform of Identity Mapping/Proof 3

== Theorem ==
== Proof ==
From Laplace Transform of Derivative:
:$(1): \quad \laptrans {\map {I_\R'} t} = s \laptrans {\map {I_\R} t} - \map {I_\R} 0$
under suitable conditions.
Then:
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Laplace Transform of Derivative: $15 \ \text{(b)}$
Category:Laplace Transform of Identity Mapping
Laplace Transform of Integral

== Theorem ==
Let $f: \R \to \R$ or $\R \to \C$ be a function.
Let $\laptrans f = F$ denote the Laplace transform of $f$.
Then:
:$\ds \laptrans {\int_0^t \map f u \rd u} = \dfrac {\map F s} s$
wherever $\laptrans f$ exists.
== Proof ==
Let $\map g t = \ds \int_0^t \map f u \rd u$.
Then:
:$\map {g'} t = \map f t$
and:
:$\map g 0 = 0$
Thus:
== Examples ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $6$. Laplace transform of integrals: Theorem $1 \text{-} 11$
* : Chapter $1$: The Laplace Transform: Solved Problems: Laplace Transform of Integrals: $17$
* : $\S 32$: Table of General Properties of Laplace Transforms: $32.13$
Category:Laplace Transforms
Laplace Transform of Integral/Examples/Example 1

== Examples of Use of Laplace Transform of Integral ==
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
:$\ds \laptrans {\int_0^1 \sin 2 u \rd u} = \dfrac 2 {s \paren {s^2 + 4} }$
== Proof ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $6$. Laplace transform of integrals
Category:Examples of Laplace Transforms
Laplace Transform of Multiple Integral

== Theorem ==
Let $f: \R \to \R$ or $\R \to \C$ be a function.
Let $\laptrans f = F$ denote the Laplace transform of $f$.
Then for all $n \in \Z_{\ge 0}$:
:$\ds \laptrans {\underbrace {\int_0^t \dotsm \int_0^t}_{\text {$n$ times} } \map f u \rd u^n} = \dfrac {\map F s} {s^n}$
wherever $\laptrans f$ exists.
== Proof ==
The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\ds \laptrans {\underbrace {\int_0^t \dotsm \int_0^t}_{\text {$n$ times} } \map f u \rd u^n} = \dfrac {\map F s} {s^n}$
$\map P 0$ is the case:
:$\map f u = \map F s$
which is the statement of the Laplace transform.
Thus $\map P 0$ is seen to hold.
=== Basis for the Induction ===
$\map P 1$ is the case:
:$\ds \laptrans {\int_0^t \map f u \rd u} = \dfrac {\map F s} s$
which is established in Laplace Transform of Integral
Thus $\map P 1$ is seen to hold.
This is the basis for the induction.
=== Induction Hypothesis ===
Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
:$\ds \laptrans {\underbrace {\int_0^t \dotsm \int_0^t}_{\text {$k$ times} } \map f u \rd u^k} = \dfrac {\map F s} {s^k}$
from which it is to be shown that:
:$\ds \laptrans {\underbrace {\int_0^t \dotsm \int_0^t}_{\text {$k + 1$ times} } \map f u \rd u^{k + 1} } = \dfrac {\map F s} {s^{k + 1} }$
=== Induction Step ===
This is the induction step:
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
:$\forall n \in \Z_{\ge 0}: \ds \laptrans {\underbrace {\int_0^t \dotsm \int_0^t}_{\text {$n$ times} } \map f u \rd u^n} = \dfrac {\map F s} {s^n}$
== Sources ==
* : $\S 32$: Table of General Properties of Laplace Transforms: $32.14$
Category:Laplace Transforms
Laplace Transform of Natural Logarithm

== Theorem ==
:$\laptrans {\ln t} = \dfrac {\map {\Gamma'} 1 - \ln s} s = -\dfrac {\gamma + \ln s} s$
where:
:$\laptrans f$ denotes the Laplace transform of the function $f$
:$\Gamma$ denotes the Gamma function
:$\gamma$ denotes the Euler-Mascheroni constant.
== Proof 1 ==
== Proof 2 ==
Category:Laplace Transforms
Category:Natural Logarithms
Category:Laplace Transform of Natural Logarithm
Laplace Transform of Natural Logarithm/Proof 2

== Theorem ==
== Proof ==
From Laplace Transform of Power:
:$\ds \int_0^\infty e^{-s t} t^k \rd t = \dfrac {\map \Gamma {k + 1} } {s^{k + 1} }$
for $k > -1$.
Differentiating $k$:
:$\ds \int_0^\infty e^{-s t} t^k \ln t \rd t = \dfrac {\map {\Gamma'} {k + 1} - \map \Gamma {k + 1} \ln s} {s^{k + 1} }$
Setting $k = 0$:
== Sources ==
* : Chapter $1$: The Laplace Transform: Solved Problems: Miscellaneous Problems: $50$
Category:Laplace Transform of Natural Logarithm
Laplace Transform of Null Function

== Theorem ==
Let $\NN: \R \to \R$ be a null function.
The Laplace transform of $\map \NN t$ is given by:
:$\laptrans {\map \NN t} = 0$
== Proof ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Laplace Transforms of Special Functions: $14$
Category:Laplace Transforms
Category:Null Functions
Laplace Transform of Periodic Function

== Theorem ==
Let $f: \R \to \R$ be a real function.
Let $f$ be periodic, that is:
:$\exists T \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + T}$
Then:
:$\laptrans {\map f t} = \dfrac 1 {1 - e^{-s T} } \ds \int_0^T e^{-s t} \map f t \rd t$
where $\laptrans {\map f t}$ denotes the Laplace transform.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Sources ==
* : Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $9$. Periodic functions: Theorem $1 \text{-} 14$
* : $\S 32$: Table of General Properties of Laplace Transforms: $32.17$
Category:Laplace Transforms
Category:Laplace Transform of Periodic Function