Fourier series, Fourier transform, Green's functions｜海浪晶灵aa

1. Formula
   Observation

   Write the integration variable x on the whiteboard, appearing in $\int_{0}^{\infty} \frac{\sin x}{x} dx$ and $\int_{0}^{\infty} (\frac{\sin x}{x})^2 dx$.

x

integration variable

real number

1. Formula
   Observation

   Write $f(t)$ on the whiteboard, and use |t| $\le 1$ in the piecewise condition.

t

independent variable of function f

real number

1. Formula
   Observation

   Write $F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$ on the whiteboard.

$\omega$

frequency variable of the Fourier transform

real number

1. Formula
   Observation

   Write the piecewise function $f(t) = 1$ for |t| <= 1, and 0 otherwise on the whiteboard.

$f(t)$

the rectangular pulse function, also known as the gate function, equals 1 when the absolute value of t is less than or equal to 1 and equals zero elsewhere.0

t is a real number.

1. Formula
   Observation

   The variable $\omega$ appears in the Fourier transform formula.

$\omega$

Angular frequency variable.

$\omega$ is a real number.

1. Formula
   Observation

   Write $F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$ on the whiteboard.

$F(\omega)$

Fourier transform of the function $f(t)$.

$\omega$ is a real number.

1. Formula
   Observation

   The whiteboard shows $f(t) = 1$ (|t| ≤ 1), $f(t) = 0$ (otherwise), accompanied by a rectangular pulse graph.

$f(t)$

a rectangular pulse function of the time variable t, which takes the value 1 when |t| is less than or equal to 1, and takes the value 0 elsewhere.

t is a real number.

1. Formula
   Observation

   The variable ω appears in both the Fourier transform and the inverse Fourier transform formulas.

$\omega$

The angular frequency variable, serving as the independent variable after Fourier transformation.

ω is a real number.

1. Formula
   Observation

   The whiteboard displays $F(ω) = \int _{-\infty }^{\infty } f(t) e^{-iωt} dt$, and subsequent substitution yields $2 \sin ω / ω$.

$F(\omega)$

The Fourier transform of $f(t)$.

ω is a real number.

1. Formula
   Observation

   The exponential factors are written as $e^{-iωt}$ and $e^{iωt}$.

i

The imaginary unit.

The square of i equals negative one.

1. Formula
   Observation

   Both the target integral at the top and the conclusion at the bottom involve π and π divided by 2.

$\pi$

the constant pi

a real constant

1. Formula
   Observation

   The inverse transform formula is written as $\int$_{-}^{$\infty$} $F(ω)e^{iωt} dω$.

u

The differential variable used for the inverse transform integral on the whiteboard visually resembles ω.

integration variable

1. Audio
   Observation

   The instructor says, "Today we will discuss this: the Dirichlet integral."

2. Formula
   Observation

   Write $\int_{0}^{\infty} \frac{\sin x}{x} dx = \int_{0}^{\infty} (\frac{\sin x}{x})^2 dx = \frac{\pi}{2}$ on the whiteboard.

The video first presents two improper integrals and states that they yield the same result, equal to $\frac{\pi}{2}$.

1. The lower limit of integration is 0, and the upper limit is $\infty$.

2. The integrands are $\frac{\sin x}{x}$ and $\left(\frac{\sin x}{x}\right)^{2}$, respectively.

1. Audio
   Observation

   The instructor says, "Our first method uses Fourier... and our second method solves it using complex analysis, specifically contour integration."

2. Formula
   Observation

   The whiteboard displays "1. Fourier" and "2. Complex analysis".

The video lists two methods for evaluating this integral: the first uses Fourier transforms, and the second uses contour integration from complex analysis.

1. the Dirichlet integral and its squared form

1. Audio
   Observation

   The instructor says, "Using Fourier, we need a function such that when the absolute value of t is less than or equal to 1, it equals 1; otherwise, in all other cases, it equals 0."

2. Diagram
   Observation

   The whiteboard shows a rectangular pulse graph, with the horizontal axis labeled -1 and 1, and the vertical axis labeled 1.

3. Formula
   Observation

   The whiteboard displays $f(t) = \begin{cases} 1 & (|t| \le 1) \\ 0 & (\text{other}) \end{cases}$.

To use the Fourier method, the video introduces a piecewise function that takes the value 1 to 1 on the interval [1, and takes the value 0 elsewhere.

1. When |t| ≤ 1, $f(t) = 1$

2. When |t| > 1, $f(t) = 0$

1. Formula
   Observation

   Write the definition of the Fourier transform, $F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$, on the whiteboard.

The Fourier transform of the function $f(t)$ is defined as $F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$.

1. The integral converges

1. Formula
   Observation

   Write the definition of the inverse Fourier transform on the whiteboard: $f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega$

The inverse Fourier transform of the function $F(\omega)$ is defined as $f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega$.

1. The integral converges

1. Definition of the Fourier transform.

1. Formula
   Observation

   Write the piecewise function on the whiteboard: $f(t) = 1$ (|t| <= 1), 0 (otherwise)

2. Diagram
   Observation

   The whiteboard shows a graph of this function: a rectangular pulse

The rectangular pulse function f of t is defined as: when the absolute value of t is less than or equal to 1, f of t equals 1; when the absolute value of t is greater than 1, f of t equals 0.

1. Audio
   Observation

   The instructor mentions 'using Euler's formula'

2. Formula
   Observation

   Expand $e^{-i\omega t}$ on the whiteboard as $\cos \omega t - i \sin \omega t$

Euler's formula establishes a connection between the complex exponential function and trigonometric functions: e^{ix} = $\cos x + i \sin x$. Here it is used to expand $e^{-i\omega t}$.

1. Audio
   Observation

   The instructor says, 'Because this is an even function... this is an odd function... so the odd function vanishes.'

2. Formula
   Observation

   On the whiteboard, the integration interval changes from [-1, 1] to 2 times [0, 1], and the sine term is removed.

Over a symmetric interval [-a, a], the integral of an odd function is 0, and the integral of an even function is twice its integral over [0, a].

1. The integration interval is symmetric about the origin.

1. Formula
   Observation

   On the whiteboard, $F(ω) = \int _{-\infty }^{\infty } f(t)e^{-iωt} dt$ is written.

In the video, the Fourier transform of the rectangular pulse $f(t)$ is defined as the integral over the entire real line: $\int _{-\infty }^{\infty } f(t)e^{-iωt} dt$.

1. $f(t)$ is the rectangular pulse function given on the whiteboard

2. The integration variable is t

3. The result is a function of ω

1. Formula
   Observation

   Write $f(t) = 1/(2π) \int _{-\infty }^{\infty } F(ω)e^{iωt}$ du on the whiteboard.

1. The differential variable is written as du, but in context it should be dω.

The video uses the inverse transform formula to express $f(t)$ as the integral of $F(ω)$ multiplied by $e^{iωt}$, then divided by 2π.

1. $F(ω)$ is the Fourier transform of $f(t)$

2. t is a real variable

1. Fourier transform definition

1. Formula
   Observation

   The whiteboard shows $f(t) = 1$ (|t| ≤ 1), $f(t) = 0$ (other).

2. Diagram
   Observation

   On the left, draw a rectangle of height 1 and base from -1 to 1.

The example function chosen in this video is a rectangular pulse that is constantly 1 to 1 and zero outside that interval.1 on the interval [0]

1. Defined on the entire real line

1. Formula
   Observation

   At the top of the whiteboard: $\int _0^\infty$ sinx/x $dx = \int _0^\infty$ (sinx/x)^$2 dx = π/2$.

The target equation was given at the start of the video; this segment derives $\int _0^\infty$ sinx/x $dx = π/2$ via Fourier transform.

1. The integrand is understood in the limit sense at $x=0$.

1. Formula
   Observation

   The whiteboard shows ∴ $π=\int _{-\infty }^{\infty } \sin ω/ω dω$.

2. Audio
   Observation

   The instructor says, "This integral equals π."

The integral over ω from negative infinity to positive infinity of sine ω divided by ω with respect to ω equals π.

1. We reuse the earlier result $F(ω)=2 \sin ω/ω$ obtained from the rectangular pulse.

2. Substitute $t=0$ into the inverse transform formula.

The improper integral over all real ω.

1. Formula
   Observation

   The whiteboard shows ∴ $π/2=\int _0^\infty \sin ω/ω dω$.

2. Audio
   Observation

   The instructor says, "Because an odd function is actually an even function," then gives the final result as π over two.

1. The audio says 'an odd function is actually an even function', which is ambiguous, but the on-screen conclusion clearly uses the evenness of $\sin ω/ω$.

The integral from zero to positive infinity of sine ω over ω with respect to ω equals one-half π.

1. The integral from negative infinity to positive infinity of sine ω over ω with respect to ω equals π.

2. $\sin ω/ω$ is an even function.

The improper integral for $ω \ge 0$

1. Audio
   Observation

   The instructor says, “Now, if we consider the Fourier transform, let’s first look at the Fourier transform of this function… It equals one only between negative one and one, so essentially from negative one to one.”

2. Formula
   Observation

   The whiteboard shows $F(ω) = \int _{-\infty }^{\infty } f(t) e^{-iωt} dt = \int _{-1}^{1} e^{-iωt} dt$.

1. Expression

   $$F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$$
   Explanation

   Write down the definition of the Fourier transform.

   Justification

   Definition of the Fourier transform

   Shown in the video
2. Expression

   $$= \int_{-1}^{1} e^{-i\omega t} dt$$
   Explanation

   Since $f(t)$ is nonzero only on [-1, 1] and equals 1 there, the integration interval is reduced and $f(t) = 1$ is substituted.

   Justification

   Piecewise definition of $f(t)$

   Shown in the video

Simplify the Fourier transform integral over the infinite interval into an integral over the finite interval [−1, 1].

1. Formula
   Observation

   The whiteboard step-by-step writes out the computation process of the Fourier transform.

1. Expression

   $$F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$$
   Explanation

   Write the integral expression according to the definition of the Fourier transform.

   Justification

   Definition of the Fourier transform.

   Shown in the video
2. Expression

   $$= \int_{-1}^{1} e^{-i\omega t} dt$$
   Explanation

   Substitute the explicit form of the rectangular pulse function $f(t)$, shrinking the integration interval to [-1, 1].

   Justification

   Definition of the rectangular pulse function.

   Shown in the video
3. Expression

   $$= \int_{-1}^{1} (\cos \omega t - i \sin \omega t) dt$$
   Explanation

   Use Euler's formula to expand the complex exponential function into trigonometric functions.

   Justification

   Euler's formula

   Shown in the video
4. Expression

   $$= 2 \int_{0}^{1} \cos \omega t dt$$
   Explanation

   Since $\cos \omega t$ is an even function and $\sin \omega t$ is an odd function, the integral of an odd function over the symmetric interval [-1, 1] is 0, and the integral of an even function over that interval is twice its integral over [0, 1].

   Justification

   Integration properties of even and odd functions

   Shown in the video
5. Expression

   $$= 2 \left[ \frac{\sin \omega t}{\omega} \right]_{0}^{1}$$
   Explanation

   Evaluate the definite integral by finding the antiderivative.

   Justification

   The Fundamental Theorem of Calculus

   Shown in the video
6. Expression

   $$= 2 \frac{\sin \omega}{\omega}$$
   Explanation

   Substitute the upper and lower limits of integration to obtain the final result.

   Justification

   Algebraic manipulation

   Shown in the video

The Fourier transform of the rectangular pulse function $f(t)$ is $F(\omega) = 2 \frac{\sin \omega}{\omega}$.

1. Formula
   Observation

   The Fourier inverse transform formula is written on the whiteboard, ready for substitution with $t = 0$

1. Expression

   $$f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega$$
   Explanation

   Write the integral expression according to the definition of the Fourier inverse transform.

   Justification

   Definition of the inverse Fourier transform

   Shown in the video
2. Expression

   $$t = 0 \Rightarrow f(0) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) d\omega$$
   Explanation

   Let $t = 0$; then $e^{i\omega t} = 1$, and $f(0) = 1$.

   Justification

   Substituting specific values

   Shown in the video

By evaluating the inverse Fourier transform at $t = 0$, the Fourier transform of the rectangular pulse function can be linked to the Dirichlet integral.

1. Formula
   Observation

   On the whiteboard, $F(ω) = \int _{-\infty }^{\infty } f(t)e^{-iωt}dt = \int _{-1}^{1} e^{-iωt}dt = \int _{-1}^{1}(\cos ωt+i \sin ωt)dt = 2\int _0^1 \cos ωt dt = 2[\sin ωt/ω]_0^1 = 2 \sin ω/ω$.

2. Audio
   Observation

   The instructor explains simultaneously that $f(t)$ is substituted and the limits of integration are reduced to -1 to 1.

1. Expression

   $$F(\omega)=\int_{-\infty}^{\infty} f(t)e^{-i\omega t}\,dt$$
   Explanation

   Starting from the definition of the Fourier transform.

   Justification

   Using the definition formula given on the whiteboard.

   Shown in the video
2. Expression

   $$=\int_{-1}^{1} e^{-i\omega t}\,dt$$
   Explanation

   Since $f(t)$ equals 1 only when |t| is less than or equal to 1, and equals zero elsewhere,0, so the integration interval shrinks to [1, 1].

   Justification

   Defined using a rectangular pulse.

   Shown in the video
3. Expression

   $$=\int_{-1}^{1}(\cos\omega t+i\sin\omega t)\,dt$$
   Explanation

   Expand the complex exponential into cosine plus i times sine.

   Justification

   Euler's formula: $e^{-iωt} = \cos ωt - i \sin ωt$; on the whiteboard it is written as $\cos ωt + i \sin ωt$, and only the cosine term contribution is retained thereafter.

   Shown in the video
4. Expression

   $$=2\int_{0}^{1}\cos\omega t\,dt$$
   Explanation

   The sine term integrates to 0 over a symmetric interval; the cosine term is an even function, so the integral reduces to 2 times the integral over half the interval.

   Justification

   Use the fact that cos is even, sin is odd, and the property of integration over a symmetric interval.

   Shown in the video
5. Expression

   $$=2\left[\frac{\sin\omega t}{\omega}\right]_{0}^{1}=2\frac{\sin\omega}{\omega}$$
   Explanation

   Integrate cos ωt with respect to t to obtain $\sin ωt / ω$, then substitute the limits of integration.

   Justification

   Basic integration formula and substitution of endpoints.

   Shown in the video

$F(ω)$ equals two times sine ω divided by ω.

1. Formula
   Observation

   On the whiteboard: $f(0) = 1 = 1/(2π) \int _{-\infty }^{\infty } 2 \sin ω/ω dω$.

2. Audio
   Observation

   The instructor says, 'That is, when t equals 0, substitute t equals 0, and $f(0)$ equals 1.'

1. Expression

   $$f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)e^{i\omega t}\,d\omega$$
   Explanation

   First write down the Fourier inverse transform formula.

   Justification

   Use the inverse transform given on the whiteboard.

   Shown in the video
2. Expression

   $$f(0)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)\,d\omega$$
   Explanation

   Let $t = 0$, then $e^{iωt} = e^0 = 1$.

   Justification

   The exponential function equals 0 at 1.

   Shown in the video
3. Expression

   $$1=\frac{1}{2\pi}\int_{-\infty}^{\infty}2\frac{\sin\omega}{\omega}\,d\omega$$
   Explanation

   The left-hand side, defined by the rectangular pulse, gives $f(0) = 1$; the right-hand side substitutes the previously derived $F(ω) = 2 \sin ω/ω$.

   Justification

   Use the definition of $f(t)$ and the already-derived Fourier transform result.

   Shown in the video

We obtain $1 = (1/(2π)) \int _{-\infty }^{\infty } 2 \sin ω/ω dω$.

1. Formula
   Observation

   The whiteboard writes: ∴ $π = \int _{-\infty }^{\infty } \sin ω/ω dω$.

2. Audio
   Observation

   The instructor says, 'This 2 and 2 cancel out,' then moves π to the other side.

1. Expression

   $$1=\frac{1}{2\pi}\int_{-\infty}^{\infty}2\frac{\sin\omega}{\omega}\,d\omega$$
   Explanation

   Start from the previous step's equation.

   Justification

   Substitute the result from the previous step.

   Shown in the video
2. Expression

   $$1=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{\sin\omega}{\omega}\,d\omega$$
   Explanation

   The 2 in the numerator cancels with the 2 in the denominator pi 2.

   Justification

   Algebraic simplification.

   Shown in the video
3. Expression

   $$\pi=\int_{-\infty}^{\infty}\frac{\sin\omega}{\omega}\,d\omega$$
   Explanation

   Multiply both sides by π.

   Justification

   Multiply both sides of the equation by a nonzero constant.

   Shown in the video

The integral from negative infinity to positive infinity of sine ω over ω with respect to ω equals π.

1. Formula
   Observation

   Write on the whiteboard: ∴ $π/2 = \int _0^\infty \sin ω/ω dω$.

2. Audio
   Observation

   The presenter says, 'Because an odd function is actually an even function,' then gives the final result as π over two.

1. The audio statement is not rigorous, but the on-screen conclusion matches the standard even-function half-axis reduction.

1. Expression

   $$\int_{-\infty}^{\infty}\frac{\sin\omega}{\omega}\,d\omega=\pi$$
   Explanation

   First cite the full-axis integral result.

   Justification

   From the previous derivation.

   Shown in the video
2. Expression

   $$\frac{\sin\omega}{\omega}\text{is an even function}$$
   Explanation

   Because $\sin (-ω)/(-ω) = \sin ω/ω$.

   Justification

   Sine is an odd function, and the denominator ω is also an odd function; their quotient is an even function.

   Derived from the video
3. Expression

   $$\int_{-\infty}^{\infty}\frac{\sin\omega}{\omega}\,d\omega=2\int_{0}^{\infty}\frac{\sin\omega}{\omega}\,d\omega$$
   Explanation

   The integral of an even function over the entire real line equals twice its integral over the half-line.

   Justification

   Symmetry property of integrals of even functions.

   Derived from the video
4. Expression

   $$\int_{0}^{\infty}\frac{\sin\omega}{\omega}\,d\omega=\frac{\pi}{2}$$
   Explanation

   Substitute the full-line integral value π and divide by 2.

   Justification

   Algebraic solution.

   Shown in the video

The integral from zero to positive infinity of sine ω over ω with respect to ω equals one-half π.

1. Formula
   Observation

   The entire whiteboard centers on $f(t) = 1$ (|t| ≤ 1), 0 (other), and ultimately writes $π/2 = \int _0^\infty \sin ω/ω dω$.

2. Audio
   Observation

   The presenter substitutes $t = 0$ step by step, cancels constants, and obtains the final result.

Given $f(t) = 1$ when |t| ≤ 1 and 0 otherwise, use the Fourier transform and its inverse to evaluate $\int _0^\infty$ sinx/x dx.

1. $f(t)$ equals one when the absolute value of t is less than or equal to one; otherwise, it equals zero.

2. $F(ω)$ equals the integral from negative infinity to positive infinity of $f(t)$ times e to the power of negative i ω t with respect to t.

3. $f(t)$ equals one over two π times the integral from negative infinity to positive infinity of $F(ω)$ times e to the power of i ω t with respect to ω.

Compute the integral $\int _0^\infty \sin x / x dx$.

1. Expression

   $$F(\omega)=\int_{-1}^{1}e^{-i\omega t}dt=2\int_0^1\cos\omega t\,dt=2\frac{\sin\omega}{\omega}$$
   Explanation

   First compute the Fourier transform of a rectangular pulse.

   Justification

   Simplify using the piecewise definition of $f(t)$ and integration over a symmetric interval.

   Shown in the video
2. Expression

   $$f(0)=1=\frac{1}{2\pi}\int_{-\infty}^{\infty}2\frac{\sin\omega}{\omega}d\omega$$
   Explanation

   In the inverse transform, set $t = 0$.

   Justification

   $f(0) = 1$ and $e^{iω\cdot 0} = 1$.

   Shown in the video
3. Expression

   $$\pi=\int_{-\infty}^{\infty}\frac{\sin\omega}{\omega}d\omega$$
   Explanation

   Cancel the constant 2 and rearrange terms.

   Justification

   Algebraic simplification.

   Shown in the video
4. Expression

   $$\frac{\pi}{2}=\int_0^{\infty}\frac{\sin\omega}{\omega}d\omega$$
   Explanation

   Use the evenness of $\sin ω / ω$ to convert the full-axis integral into a half-axis integral.

   Justification

   Even-function symmetric-integration property.

   Shown in the video

The integral from zero to positive infinity of sine x over x with respect to x equals one-half pi.

This matches the target equation prewritten at the top of the whiteboard: $\int _0^\infty \sin x / x dx = π / 2$.

1. Audio
   Observation

   The instructor says, “Using complex analysis, we compute this using a classic contour integral with a keyhole indentation.”

2. Diagram
   Observation

   The whiteboard draws the complex plane, including a large semicircle in the upper half-plane and a small semicircular indentation around the origin.

1. Complex plane coordinate axes

2. Large semicircle in the upper half-plane

3. Small semicircle at the origin

1. The instructor draws the large circular arc

2. The instructor draws a small circular arc at the origin to avoid the singularity

1. The coordinate axes remain unchanged

This sketch shows the contour structure commonly used to avoid the origin when evaluating improper integrals along the real axis via the residue theorem.

1. Diagram
   Observation

   A graph of the rectangular pulse function is drawn on the whiteboard, with the horizontal axis labeled t and the vertical axis labeled f of t; the height is 1 over the interval from negative 1 to 1

1. Coordinate axes

2. Rectangle

1. Width and height of the rectangle

Intuitively shows the shape and range of values of the integrand $f(t)$, explaining why the integration interval can be reduced from (-\infty, $\infty$) to [-1, 1].

1. Diagram
   Observation

   At the top of the whiteboard, the target integral is written first; in the middle, '1. Fourier', a rectangular diagram, and the piecewise definition of $f(t)$; on the right, the derivation of the Fourier transform and its inverse; and at the bottom, the final conclusion is filled in step by step.

1. Target integral at the top

2. Rectangular pulse graph and definition of $f(t)$ on the left

3. Derivation of $F(ω)$ in the center

4. Inverse transform and final conclusion at the bottom

1. The instructor first supplements the substitution expression $f(0)=1$ on the right

2. Then writes $π=\int _{-\infty }^{\infty } \sin ω/ω dω$ below it

3. Finally writes $π/2=\int _0^\infty \sin ω/ω dω$

1. The target equation at the top remains visible throughout

2. The rectangular pulse definition on the left remains visible throughout

The visual layout organizes 'target integral—example function—Fourier transform—substitution of $t=0$ into the inverse transform—simplification to conclusion' into a continuous chalkboard proof chain.

1. Diagram
   Observation

   In the coordinate graph on the left, draw a rectangle of height 1 spanning from -1 to 1.

1. horizontal axis

2. vertical axis

3. rectangle of height 1

4. labeled -1 and 1

1. the rectangle is nonzero only for |t| ≤ 1

2. height is constantly 1

this figure intuitively shows that $f(t)$ is a gate function of width 2 and height 1, so its Fourier transform integral limits can be reduced to [-1, 1].

1. Audio
   Observation

   the presenter says, while explaining half-axis reduction, "because an odd function is actually an even function."

2. Formula
   Observation

   the on-screen conclusion uses $\int _{-\infty }^{\infty } \sin ω/ω dω = 2\int _0^\infty \sin ω/ω dω$.

1. the audio wording is ambiguous, possibly due to a slip of the tongue or omitted explanation.

It is easy to mistakenly say "$\sin ω/ω$ is also an odd function" instead of "sin is an odd function".

sinω is an odd function, and ω is also an odd function, but their quotient $\sin ω/ω$ is an even function; the final step in this segment uses this evenness to convert the integral over the entire axis into twice the integral over the half-axis.

1. Audio
   Observation

   After writing down the integral, the instructor immediately lists two methods.

Fourier transform and complex contour integration are two methods used to compute the Dirichlet integral and its square form.

1. Audio
   Observation

   The instructor says, 'Using Fourier’s approach, we need a function like this,' then defines $f(t)$ and computes its Fourier transform.

The definition of the rectangular pulse function $f(t)$ is the prerequisite for subsequent Fourier transform calculations.

1. Audio
   Observation

   The instructor says, "This is the function we obtained after applying the Fourier transform; now we use the inverse Fourier transform..."

By computing the Fourier transform of the rectangular pulse function and then using the property of the inverse Fourier transform at a specific point ($t = 0$), the Dirichlet integral is evaluated.

1. Formula
   Observation

   The piecewise definition of $f(t)$ directly yields $F(ω) = 2 \sin ω/ω$.

The rectangular pulse definition is substituted into the Fourier transform definition, yielding a specific sinc-type expression.

1. Formula
   Observation

   First use the expression for $F(ω)$, then substitute into $f(t) = 1/(2π) \int F(ω) e^{iωt} dω$.

We must first compute $F(ω)$ before substituting it into the inverse transform and setting $t = 0$ to obtain the integral identity.

1. Formula
   Observation

   Starting from $π = \int _{-\infty }^{\infty } \sin ω/ω dω$, we further write $π/2 = \int _{0}^{\infty } \sin ω/ω dω$.

Using the evenness of the integrand, the full-axis integral result is specialized to a half-axis integral result.

1. Formula
   Observation

   The bottom conclusion matches the top target integral in form, differing only in the variable name: ω is replaced by x.

The $\int _{0}^{\infty } \sin ω/ω dω = π/2$ derived in this segment is precisely the first Dirichlet integral appearing in the top target.

1. Formula
   Observation

   Write the integral result on the whiteboard as equal to $\frac{\pi}{2}$.

1. the Dirichlet integral and its squared form

1. Formula
   Observation

   Write the piecewise definition of $f(t)$ on the whiteboard.

1. Rectangular pulse function $f(t)$.

1. Formula
   Observation

   The complete derivation is fully displayed on the whiteboard.

1. Definition of the Fourier transform.
2. Rectangular pulse function
3. Euler's formula
4. Integration properties of even and odd functions.

1. Audio
   Observation

   The instructor explains the upcoming strategy.

1. Definition of the inverse Fourier transform

1. Audio
   Observation

   The instructor explicitly states, "That is, when t equals 0, substitute t equals 0."

2. Formula
   Observation

   On the whiteboard, $f(0) = 1$ = ... is written.

1. Fourier inverse transform formula
2. Substitute $t = 0$ into the inverse transform to evaluate the integral.

1. Audio
   Observation

   The instructor says, 'This 2 and 2 cancel each other out.'

2. Formula
   Observation

   From $1 = (1 / (2 π)) \int 2 (\sin ω / ω) dω$ to $π = \int (\sin ω / ω) dω$.

1. Simplify to obtain the integral over the entire real line.
2. The full-axis sinc integral equals π.

1. Formula
   Observation

   Directly write the half-axis integral from the full-axis integral: $π / 2 = \int _{0}^{\infty } (\sin ω / ω) dω$.

2. Audio
   Observation

   The instructor mentions parity, but the wording is ambiguous.