PCM20210912_SICP_2.4.1_RepresentationsForComplexNumbers.jl

### A Pluto.jl notebook ###
# v0.19.12

using Markdown
using InteractiveUtils

# ╔═╡ 90aa1e17-89f8-4884-9a3b-e58acf87bc7a
using Plots

# ╔═╡ e39aed1d-58a3-40e6-8a73-27204b742f42
md"
=====================================================================================
#### SICP\_2.4.1\_RepresentationsForComplexNumbers.jl
###### file: PCM20210912\_SICP\_2.4.1\_RepresentationsForComplexNumbers.jl
###### code: Julia/Pluto.jl (1.8.2/0.19.14) by PCM *** 2022/12/09 ***

=====================================================================================

"

# ╔═╡ d87312c3-36cc-4d5d-9d3f-fd9da02bdadb
md"
#### 2.4.1.1 Definition of complex numbers
"

# ╔═╡ b4ce58d8-c130-4c4f-904d-deccef8940cf

let
	title = "Representation of z = Re(z) + Im(z) = x + yi in C"
	xlim  = (-0.5, 2.5); ylim = (-0.5, 1.2)
	z = 2 + 1im
	ϕ = atan(imag(z)/real(z))
	radius = 0.8
	circleArray = Plots.partialcircle(0, ϕ, 30, radius)
	plot([(0, 0), (2, 1)], line=:arrow, linecolor=:red, linewidth=4, title=title, xlim=xlim, ylim=ylim, legend=false)
	plot!([(2, 0), (2, 1)], linestyle=:dash, linecolor=:blue, linewidth=1)
	plot!([(0, 0), (2, 0)], linestyle=:dash, linecolor=:blue, linewidth=1)
	plot!([(0, 1), (2, 1)], linestyle=:dash, linecolor=:blue, linewidth=1)
	plot!([(0, 0), (0, 1)], linestyle=:dash, linecolor=:blue, linewidth=1)
	plot!(circleArray, linestyle=:dot, linecolor=:red, linewidth=1)
	annotate!(2.3,  1.04, ("z = x + yi", 12, :red))
	annotate!(2.25, 0.5,  ("y=Im(z)", 12))
	annotate!(1.0, -0.1,  ("x=Re(z)", 12))
	annotate!(1.15, 0.2,  ("ϕ=arctan(y/x)", 12, :red))
	annotate!(0.85, 0.6,  ("|z|=ρ=r", :red, 12))
end # let

# ╔═╡ 576d32a1-4aa4-414d-84a5-99a1f5bc3208
md"
**Fig. 2.4.1.1** Complex numbers as *points* (or *vectors*) in $$\mathbb C$$ (cf. Fig. 2.20 in SICP)

---
"

# ╔═╡ 24a94004-d2f6-40f2-9596-0d2467fe8f4b
md"

$$z = x + yi = r \cdot cos \, \phi + r \cdot i\, sin \, \phi = r \cdot (cos \, \phi + i\, sin \, \phi) = r \cdot e^{i \, \phi}$$

where:

$$x  = Re(z) = r \cdot cos \, \phi$$
$$yi = Im(z) = r \cdot i\, sin \, \phi$$
$$ϕ  = arctan(Im(z)/Re(z)) = arctan(y/x)$$
$$ρ  = r = |z| = \sqrt{x^2+y^2}$$
"

# ╔═╡ c2bfc649-b5b1-4e75-83f4-76d2aa57bc5b
md"
---
##### [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula) for complex analysis
The expression

$$e^{i \, \phi} = (cos \, \phi + i\, sin \, \phi)$$

is known as *Euler's formula*. Several different [proofs](https://en.wikipedia.org/wiki/Euler%27s_formula#Proofs) are known. One of the first was using the *power series expansion* of all terms in the formula.
"

# ╔═╡ eb2b46af-1eda-4b19-988d-ab8b2f7c3525
md"
##### [Euler's identity](https://en.wikipedia.org/wiki/Euler%27s_identity)

If we substitute $$\phi = \pi = \frac{2\pi}{2r} = \frac{circumference}{diameter}$$ *of circle* in Euler's formula:

$$e^{i \, \phi} = (cos \, \phi + i\, sin \, \phi)$$

the result is called *Euler's identity* :

$$e^{i \, \pi} = (cos \, \pi + i\, sin \, \pi) = -1 + 0i = -1$$

and rearranging terms

$$e^{i \, \pi} + 1 = 0.$$

This is known as '*our jewel*' and '*the most remarkable formula in mathematics*' [https://en.wikipedia.org/wiki/Mathematical_beauty](https://en.wikipedia.org/wiki/Mathematical_beauty) because it connects several fundamental concepts in *one* formula.

"

# ╔═╡ 46638e2b-704f-43df-9b61-a387762f78c9
let
	title = "Representation by Euler's formula in C"
	xlim  = (-1.5, 3.5); ylim = (-1.5, 1.5)
	ϕ = atan(1/2)
	radius = 1.0
	r = sqrt(2^2 + 1^2)
	z = r*exp(im*ϕ)
	circleArray = Plots.partialcircle(0, 2π, 240, radius)
	plot([(0, 0), (real(z), imag(z))], line=:arrow, linecolor=:red, linewidth=4, title=title, xlim=xlim, ylim=ylim, legend=false)
	plot!([(0, 0),   (real(z)/r, imag(z)/r)], line=:arrow, linecolor=:blue, linewidth=4)
	plot!([(2, 0),   (2, 1)], linestyle=:dash, linecolor=:blue, linewidth=1)
	plot!([(0,-1.3), (0, 1.5)], linestyle=:dash, linecolor=:black, linewidth=1)
	plot!([(0, 0),   (2, 0)], linestyle=:dash, linecolor=:blue, linewidth=1)
	plot!([(-1.3, 0),(3, 0)], linestyle=:dash, linecolor=:black, linewidth=1)
	plot!([(0, 1),   (2, 1)], linestyle=:dash, linecolor=:blue, linewidth=1)
	plot!([(0, 0),   (0, 1)], linestyle=:dash, linecolor=:blue, linewidth=1)
	plot!(circleArray, linestyle=:dot, linecolor=:blue, linewidth=1)
	annotate!(2.6, 1.05, ("z = |z|⋅exp(iϕ)", 12, :red))
	annotate!(1.3, 0.3, ("exp(iϕ)", 12, :blue))
	annotate!(2.14, 0.5, ("y", 12))
	annotate!(1.2,-0.1, ("x", 12))
	annotate!(0.6, 0.15, ("ϕ", 12, :blue))
	annotate!(1.2, 0.8, ("|z|", 12, :red))
end # let

# ╔═╡ 724ba211-8069-43b8-99b1-55d755ed3c68
md"
**Fig. 2.4.1.2** Complex numbers as *points* (or *vectors*) in $$\mathbb C$$ using *Euler's formula*

---
"

# ╔═╡ d0ed924a-378d-48d3-b0f9-ffd98ef98208
function eulersFormula(z::Complex)
	(abs(z), angle(z), abs(z)*exp(im*angle(z)))
end # function eulersFormula

# ╔═╡ 0af14617-878d-445f-a230-fdfc04d0d5fb
md"
[Complex(2, 1)](https://www.intmath.com/complex-numbers/convert-polar-rectangular-interactive.php) ==> 2.00 + 1.00i = 2.24 ∠ 0.46 radians
"

# ╔═╡ 1aa0467e-da7c-4253-b2dd-8e41931337b6
eulersFormula(Complex(2, 1))

# ╔═╡ afd5a931-b8ad-46f8-b0f2-894d59d53959
let
	title = "Representation of Euler's identity in C"
	xlim  = (-2.5, 3.5); ylim = (-1.5, 1.5)
	ϕ = π
	radius = 1.0
	r = 1
	z = r*exp(im*π)
	circleArray = Plots.partialcircle(0, 2π, 240, radius)
	plot([(0, 0), (real(z), imag(z))], line=:arrow, linecolor=:red, linewidth=4, title=title, xlim=xlim, ylim=ylim, legend=false, aspect_ratio=:equal)
	plot!([(0,-1.3), (0, 1.5)], linestyle=:dash, linecolor=:black, linewidth=1)
	plot!([(-1.3, 0),(3, 0)], linestyle=:dash, linecolor=:black, linewidth=1)
	plot!(circleArray, linestyle=:dot, linecolor=:blue, linewidth=1)
	annotate!(-1.0, 0.15, ("z = exp(iπ)", 12, :red))
	annotate!(-1.0,-0.15, ("= -1+0i", 12, :red))
end # let

# ╔═╡ e791db57-a399-47fe-ac45-f4a61c493473
md"
**Fig. 2.4.1.3** Complex number as *point* (or *vector*) in $$\mathbb C$$ using *Euler's identity*
"

# ╔═╡ 262fb6be-d103-4407-b30b-ff5293ce2bea
md"
---
#### 2.4.1.2 Scheme-like Julia
"


# ╔═╡ 6d2eea9e-fd56-43c6-95a7-3d565dd0836a
md"
$$\begin{array}{|c|c|}                     
\hline                                                         \\
layer         & \text{Operations or Functions}                 \\
              &                                                \\
\hline                                                         \\
top           & \text{representation independent}              \\
              & \text{------------------------------------}    \\
(domain)      & addComplex                                     \\
              & subComplex                                     \\
              & mulComplex                                     \\
              & divComplex                                     \\
\hline                                                         \\
middle        & \text{representation dependent}                \\
(interface)   & \text{------------------------------------}    \\
              & \begin{array}{cc}                              \\
              & realPartOfZRect       & realPartOfZPolar       \\      
              & imagPartOfZRect       & imagPartOfZPolar       \\
              & magnitudeOfZRect      & magnitudeOfZPolar      \\
              & angleFromOfZRect      & angleOfZPolar          \\
              & makeZRectFromRealImag & makeZPolarFromRealImag \\
              & makeZRectFromMagAng   & makeZPolarFromMagAng   \\
              & \end{array}                                    \\
\hline                                                         \\
ground        & cons                                           \\
(Scheme-like) & car                                            \\            
              & cdr                                            \\
              &                                                \\
\hline                                                                              
\end{array}$$                                       
"

# ╔═╡ 1e2aa2f3-81a8-4252-be4f-aa575918a67b
md"
**Fig. 2.4.1.4** *Abstraction barriers* in the complex number system (cf. to SICP, Fig. 2.19)

---
"

# ╔═╡ 4dc1459c-070f-4970-9e67-c9facdc15cef
md"
---
##### *Methods* of Scheme-like *constructor* $$cons$$
"

# ╔═╡ 82578c79-a6f3-4374-8ee2-aa8a585f4b4a
struct Cons
	car
	cdr
end

# ╔═╡ 8a06336c-044c-4091-b7ae-f4ea60b97afa
cons(car::Any, cdr::Any)::Cons = Cons(car, cdr)::Cons  

# ╔═╡ ed5d2b26-d30a-4d13-8eef-6e1175699f57
function cons(car::Any, list2::Vector)::Vector
	conslist = list2
	pushfirst!(conslist, car)
	conslist
end

# ╔═╡ f1181c0f-0127-4761-a471-757f71dddb97
function cons(list1::Vector, list2::Vector)::Vector
	conslist = push!([], list1)
	for xi in list2
		push!(conslist, xi)
	end
	conslist
end

# ╔═╡ 2e5ccb79-6d54-4b84-9a78-633ed533c1dc
md"
##### *Methods* of Scheme-like *selectors* $$car, cdr$$
"

# ╔═╡ 43b5159b-40b1-41fb-8547-a889b8e9e729
car(cell::Cons) = cell.car

# ╔═╡ 5a5a7926-7733-40cd-8ab9-583a15e534ba
car(x::Vector) = x[1]

# ╔═╡ 04532459-8d67-4ea2-9b32-f010e3100932
cdr(cell::Cons)::Any = cell.cdr

# ╔═╡ 7cde6bad-2ff7-4fec-bed6-b5d00adfb0ba
cdr(x::Vector) = x[2:end]

# ╔═╡ c02043ce-730e-42e1-ae7d-96963bdb4a75
md"
---
##### *Rectangular*-dependent functions 
(Ben's representation)
"

# ╔═╡ ad27ac18-e0f8-4b63-942d-e5a24efbd168
md"
###### Selectors
$$realPartOfZRect, imagPartOfZRect$$
$$magnitudeOfZRect, angleOfZRect$$

###### Constructors
$$makeZRectFromRealImag, makeZRectFromMagAng$$

---
"

# ╔═╡ ace9faf1-33cf-4086-b6aa-9b1fad9ddd82
realPartOfZRect(z) = car(z)

# ╔═╡ 0bb28e73-6374-4b59-8070-d6964a53c20c
imagPartOfZRect(z) = cdr(z)

# ╔═╡ 6f3732d7-f9e6-4c6b-8a18-47e128337531
magnitudeOfZRect(z) = sqrt(realPartOfZRect(z)^2 + imagPartOfZRect(z)^2)

# ╔═╡ 4775c4b5-c6e0-45da-8a1c-33070e5348ed
angleOfZRect(z) = atan(imagPartOfZRect(z)/realPartOfZRect(z))

# ╔═╡ 9c4e42b6-a630-4263-95b0-40c072763410
makeZRectFromRealImag(x, y) = cons(x, y)

# ╔═╡ fe8c3cae-b495-4670-983f-fcef9d72e98f
makeZRectFromMagAng(r, a) = cons(r * cos(a), r * sin(a))

# ╔═╡ 420220d7-586f-4f26-b107-e68677d10b5b
md"
---
##### *Polar*-dependent functions
(Alyssa's representation) 

###### Selectors
$$realPartOfZPolar, imagPartOfZPolar$$
$$magnitudeOfZPolar, anglePartOfZPolar$$

###### Constructors
$$makeZPolarFromRealImag, makeZPolarFromMagAng$$

---
"

# ╔═╡ e48844ca-fa46-4c8f-9bb8-f2d33ad7fdad
magnitudeOfZPolar(z) = car(z)

# ╔═╡ 42b1742d-da96-4ea7-b5a8-288ea45d035c
angleOfZPolar(z) = cdr(z)

# ╔═╡ fd0dcca8-dd8a-416d-91d2-469f81dba264
realPartOfZPolar(z) = magnitudeOfZPolar(z) * cos(angleOfZPolar(z))

# ╔═╡ 55535d43-7c35-4ac1-9bf8-22881647651d
imagPartOfZPolar(z) = magnitudeOfZPolar(z) * sin(angleOfZPolar(z))

# ╔═╡ 60dead94-fe17-412a-8574-e7ba010090ca
makeZPolarFromRealImag(x, y) = cons(sqrt(x^2+y^2), atan(y/x))

# ╔═╡ 93dc4413-e5f8-4df8-9798-f1189962e4e7
makeZPolarFromMagAng(r, a) = cons(r, a)

# ╔═╡ 74fd815e-34fa-4f89-aa06-d19ed3ca382f
md"
---
##### Domain (representation *in*dependent) operations
$$addComplex, subComplex, mulComplex divComplex$$
"

# ╔═╡ 64035b6c-2596-4977-bafb-96b2a0a0d90b
md"
---
###### Complex addition
"

# ╔═╡ ff97e2e0-d581-470d-bdfb-92c4822e7970
md"
**Fig. 1.4.1.5** Complex *addition* as vector addition in $$\mathbb C$$

---
"

# ╔═╡ 27d313ff-84f8-44bb-ae90-9e71fed9d106
md"
###### [Addition](https://en.wikipedia.org/wiki/Complex_number#Addition_and_subtraction) in $$\mathbb C$$

$$addComplex: \mathbb C \times \mathbb C \rightarrow \mathbb C$$
$$addComplex: (z_1, z_2)=((x_1+y_1i), (x_2+y_2i) \mapsto ((x_1+x_2),(y_1+y_2)i)$$
"

# ╔═╡ 43a53b2a-0a0d-41a5-8e7c-b3106d999272
function addComplex(z1, z2)
	makeZRectFromRealImag(
		realPartOfZRect(z1) + realPartOfZRect(z2),
		imagPartOfZRect(z1) + imagPartOfZRect(z2))
end # function addComplex

# ╔═╡ 37766cb9-6453-47a5-b127-a87fb100f868
let
	z1 = makeZRectFromRealImag(2, 1)
	r1 = magnitudeOfZRect(z1)
	z2 = makeZRectFromRealImag(realPartOfZRect(z1)/r1, -2*imagPartOfZRect(z1)/r1)
	z3 = addComplex(z1, z2)
end # let

# ╔═╡ f4ee47dc-6059-4980-86a5-15b122b56751
md"
---
###### Complex subtraction
"

# ╔═╡ 4db3fda8-63e9-48ab-ba9d-431f2ac7e061
md"
**Fig. 1.4.1.6** Complex *subtraction* as vector subtraction in $$\mathbb C$$

---
"

# ╔═╡ c635a131-54c0-4530-9735-07fc558240fb
md"
###### [Subtraction](https://en.wikipedia.org/wiki/Complex_number#Addition_and_subtraction) in $$\mathbb C$$

$$subComplex: \mathbb C \times \mathbb C \rightarrow \mathbb C$$
$$subComplex: (z_1, z_2)=((x_1+y_1i), (x_2+y_2i) \mapsto ((x_1-x_2),(y_1-y_2)i)$$
"

# ╔═╡ 6c151933-886d-4774-a380-1ba185a1070d
function subComplex(z1, z2)
	makeZRectFromRealImag(
		realPartOfZRect(z1) - realPartOfZRect(z2),
		imagPartOfZRect(z1) - imagPartOfZRect(z2))
end # function subComplex

# ╔═╡ e3cb6006-4bf0-497c-8420-55637737a981
let
	z1 = makeZRectFromRealImag(2, 1)
	r1 = magnitudeOfZRect(z1)
	z2 = makeZRectFromRealImag(realPartOfZRect(z1)/r1, -2*imagPartOfZRect(z1)/r1)
	z3 = subComplex(z1, z2)
end # let

# ╔═╡ ceed4c01-d4ca-47a7-add3-b14c775d95d8
md"
---
###### Complex multiplication
"

# ╔═╡ 46f29c21-f7c6-416f-8915-a957abf74186
let
	xlim  = (-1.5, 8.5); ylim = (-1.5, 3.0)
	z1 = 2 - 1im
	ϕ1 = atan(imag(z1)/real(z1))
	r1 = abs(z1)
	z2 = 2 + 2im	
	r2 = abs(z2)
	ϕ2 = atan(imag(z2)/real(z2))
	z3 = z1 * z2
	r3 = r1 * r2
	ϕ3 = ϕ1 + ϕ2
	circleArray  = Plots.partialcircle(0, 2π, 240, 1)
	circleArray1 = Plots.partialcircle(0, ϕ1, 30, r1)	
	circleArray2 = Plots.partialcircle(0, ϕ2, 30, r2)	
	circleArray3 = Plots.partialcircle(0, ϕ3, 30, r3)	
	plot([(0, 0), (real(z1), imag(z1))], line=:arrow, linecolor=:green, linewidth=4, xlim=xlim, ylim=ylim, legend=false, aspect_ratio=:equal)
	plot!([(0, 0),  (real(z2), imag(z2))], line=:arrow, linecolor=:blue, linewidth=4)
	plot!([(0, 0),  (real(z3), imag(z3))], line=:arrow, linecolor=:red, linewidth=4)
	plot!([(0,-1.3), (0, 2.5)], linestyle=:dash, linecolor=:black, linewidth=1)
	plot!([(-1.3, 0),(6.5, 0)], linestyle=:dash, linecolor=:black, linewidth=1)
	plot!(circleArray,  linestyle=:dot, linecolor=:blue,  linewidth=1)
	plot!(circleArray1, linestyle=:dot, linecolor=:green, linewidth=1)
	plot!(circleArray2, linestyle=:dot, linecolor=:blue,  linewidth=1)
	plot!(circleArray3, linestyle=:dot, linecolor=:red,   linewidth=1)
	annotate!(2.3, -1.0, ("z1", 12, :green))
	annotate!(1.0, -0.8, ("|z1|", 12, :green))
	annotate!(2.3,  2.1, ("z2", 12, :blue))
	annotate!(1.0,  1.5, ("|z2|", 12, :blue))
	annotate!(7.0,  2.1, ("z3=z1*z2", 12, :red))
	annotate!(4.0,  2.15,("|z1|*|z2|", 12, :red))
	annotate!(4.0,  1.7, ("=|z3|", 12, :red))
	annotate!(1.9, -0.4, ("ϕ1", 12, :green) )
	annotate!(2.2,  1.2, ("ϕ2", 12, :blue))
	annotate!(5.2,  0.8, ("ϕ1+ϕ2=ϕ3", 12, :red))
end # let

# ╔═╡ 36cf2a48-6f57-4a78-a55d-cab8199eba97
md"
**Fig. 1.4.1.5** Complex *multiplication* as vector operation in $$\mathbb C$$

---
"

# ╔═╡ a064adda-624f-4dd6-9521-1dcfb2840189
md"
###### [Multiplication](https://en.wikipedia.org/wiki/Complex_number#Addition_and_subtraction) in $$\mathbb C$$
###### in *rectangular* coordinates

$$mulComplex: \mathbb C \times \mathbb C \rightarrow \mathbb C$$
$$mulComplex: (z_1, z_2) := ((x_1+y_1i), (x_2+y_2i) \mapsto (x_1x_2 + x_1y_2i + y_1ix_2 + y_1i\cdot y_2i)$$
$$=(x_1x_2 + x_1y_2i + y_1ix_2 - y_1y_2)=(x_1x_2 - y_1y_2)+(x_1y_2 + y_1x_2)i$$

###### and in *polar* coordinates

$$mulComplex: \mathbb C \times \mathbb C \rightarrow \mathbb C$$
$$mulComplex: (z_1, z_2) := ((\rho(z_1), \phi(z_1)), (\rho(z_2), \phi(z_2))) \mapsto (\rho(z_1) \cdot \rho(z_2), \phi(z_1) + \phi(z_2))$$
"



# ╔═╡ f179ef07-e391-4894-a5f2-6cef7659e592
function mulComplex(z1, z2)
	makeZRectFromMagAng(
		magnitudeOfZRect(z1) * magnitudeOfZRect(z2),
		angleOfZRect(z1) + angleOfZRect(z2))
end # function mulComplex

# ╔═╡ 25151974-8b0b-4e33-ac6c-927e695789d0
let
	z1 = makeZRectFromRealImag(2, -1)
	z2 = makeZRectFromRealImag(2,  2)
	z3 = mulComplex(z1, z2)
end # let

# ╔═╡ 1fb809a4-7218-41c3-a54d-b524ab8192f9
md"
---
###### Complex division
"

# ╔═╡ 72e9c6eb-6404-4fc8-bafa-996d335734fe
let
	xlim  = (-2.5, 4.5); ylim = (-2.5, 3.0)
	z1 = 3 - 1im
	ϕ1 = atan(imag(z1)/real(z1))
	r1 = abs(z1)
	z2 = 1.0 + 2im	
	r2 = abs(z2)
	ϕ2 = atan(imag(z2)/real(z2))
	z3 = z1 / z2
	r3 = r1 / r2
	ϕ3 = ϕ1 - ϕ2
	circleArray  = Plots.partialcircle(0, 2π, 240, 1)
	circleArray1 = Plots.partialcircle(0, ϕ1, 30, r1)	
	circleArray2 = Plots.partialcircle(0, ϕ2, 30, r2)	
	circleArray3 = Plots.partialcircle(0, ϕ3, 30, r3)	
	plot([(0, 0), (real(z1), imag(z1))], line=:arrow, linecolor=:green, linewidth=4, xlim=xlim, ylim=ylim, legend=false, aspect_ratio=:equal)
	plot!([(0, 0),  (real(z2), imag(z2))], line=:arrow, linecolor=:blue, linewidth=4)
	plot!([(0, 0),  (real(z3), imag(z3))], line=:arrow, linecolor=:red, linewidth=4)
	plot!([(0,-1.3), (0, 2.5)], linestyle=:dash, linecolor=:black, linewidth=1)
	plot!([(-2.0, 0),(6.5, 0)], linestyle=:dash, linecolor=:black, linewidth=1)
	plot!(circleArray,  linestyle=:dot, linecolor=:blue,  linewidth=1)
	plot!(circleArray1, linestyle=:dot, linecolor=:green, linewidth=1)
	plot!(circleArray2, linestyle=:dot, linecolor=:blue,  linewidth=1)
	plot!(circleArray3, linestyle=:dot, linecolor=:red,   linewidth=1)
	annotate!(3.2, -1.0, ("z1", 12, :green))
	annotate!(1.9, -0.4, ("|z1|", 12, :green))
	annotate!(1.0,  2.2, ("z2", 12, :blue))
	annotate!(0.4,  1.5, ("|z2|", 12, :blue))
	annotate!(-.4, -1.6, ("z1/z2=z3", 12, :red))
	annotate!(-1.0, -.7, ("|z1|/|z2|=|z3|", 12, :red))
	annotate!(2.9, -0.4, ("ϕ1", 12, :green))
	annotate!(1.7,  1.0, ("ϕ2", 12, :blue))
	annotate!(1.6, -1.1, ("ϕ3=ϕ1-ϕ2", 12, :red))
end # let

# ╔═╡ f799d74f-5629-4171-86f1-d1ef772a617b
md"
###### [Division](https://en.wikipedia.org/wiki/Complex_number#Addition_and_subtraction) in $$\mathbb C$$
###### in *rectangular* coordinates

$$divComplex: \mathbb C \times \mathbb C \rightarrow \mathbb C$$
$$divComplex: (z_1, z_2) := \frac{z_1}{z_2} = \frac{z_1}{z_2}\times\frac{\overline{z_2}}{\overline{z_2}}=\frac{z_1\overline{z_2}}{z_2\overline{z_2}}=\frac{Re(z_1\overline{z_2})}{|z_2|^2}+\frac{Im(z_1\overline{z_2})}{|z_2|^2}i$$

or more explicit:

$$\frac{x_1 + y_1i}{x_2 + y_2i} = \frac{x_1 + y_1i}{x_2 + y_2i} \times \frac{x_2 - y_2i}{x_2 - y_2i}=\frac{(x_1 + y_1i)(x_2 - y_2i)}{(x_2 + y_2i)(x_2 - y_2i)}$$

$$=\frac{(x_1 x_2 + y_1 y_2) + (x_2 y_1 - x_1 y_2)i}{x_2^2 + y_2^2}=\frac{x_1 x_2 + y_1 y_2}{x_2^2 + y_2^2} + \frac{x_2 y_1 - x_1 y_2}{x_2^2 + y_2^2}i.$$

###### and in *polar* coordinates

$$mulComplex: \mathbb C \times \mathbb C \rightarrow \mathbb C$$
$$mulComplex: (z_1, z_2) := ((\rho(z_1), \phi(z_1)), (\rho(z_2), \phi(z_2))) \mapsto (\rho(z_1) / \rho(z_2), \phi(z_1) - \phi(z_2))$$
"

# ╔═╡ 0c54eddf-4cd1-4428-9d01-77742cc46796
function divComplex(z1, z2)
	makeZRectFromMagAng(
		magnitudeOfZRect(z1) / magnitudeOfZRect(z2),
		angleOfZRect(z1) - angleOfZRect(z2))
end # function divComplex

# ╔═╡ e86b123d-5658-4caa-9826-f400beda75ef
let
	z1 = makeZRectFromRealImag(3, -1)
	z2 = makeZRectFromRealImag(1,  2)
	z3 = divComplex(z1, z2)
end # let

# ╔═╡ a4a8db4c-f63f-49f5-8ded-753d55a9d8ad
md"
---
###### Inverse of complex number
$$z^{-1} = \frac{1}{z} = \frac{1}{z}\times\frac{\overline{z}}{\overline{z}} = \frac{\overline{z}}{z\overline{z}} = \frac{Re(\overline{z})}{|z|^2} + \frac{Im(\overline{z})}{|z|^2}$$ 

or more explicit:

$$\frac{1}{x+yi}=\frac{1}{x+yi} \times \frac{x-yi}{x-yi}=\frac{x-yi}{x^2+y^2}$$
$$=\frac{x}{x^2+y^2}-\frac{y}{x^2+y^2}i.$$
"

# ╔═╡ 47de4dbc-7f48-42cd-a4ed-0ef524262fd8
let
	z1 = makeZRectFromRealImag(1,  0)  # 1 + 0i = 1
	z2 = makeZRectFromRealImag(1,  2)  # 1 + 2i
	z3 = divComplex(z1, z2)            # 1/(1 + 2i) = 1/5 + 2/5i = 0.2 + 0.4i
end # let      

# ╔═╡ 78c508b3-40e1-45fd-80b3-70174ea7b5b3
md"
---
#### 2.3.1.3 idiographic Julia
"

# ╔═╡ 8d659f59-ba30-4960-a755-866b08f270a0
md"
##### Example
"

# ╔═╡ 795ca456-6ca6-4d11-b2fc-8a9014a653ad
md"
###### Definition of $$z$$
"

# ╔═╡ ed0dd90d-253b-45ab-a6c1-3bc25c5aeac9
md"
$$z = x + yi = 2 + 1i$$
"

# ╔═╡ 2c6136a7-aaea-4f34-8cf0-e6c59051c669
z = 2 + 1im

# ╔═╡ b818345b-d79b-41a4-8b6c-1e88248a2089
let
	xlim  = (-1.5, 4.5); ylim = (-1.5, 1.5)
	ϕ = atan(1/2)
	radius = 1.0
	r = sqrt(2^2 + 1^2)
	z1 = r*exp(im*ϕ)
	z2 = real(z)/r + -2*imag(z)/r*im
	z3 = z1 + z2
	circleArray = Plots.partialcircle(0, 2π, 240, radius)
	plot([(0, 0), (real(z1), imag(z1))], line=:arrow, linecolor=:green, linewidth=4, xlim=xlim, ylim=ylim, legend=false, aspect_ratio=:equal)
	plot!([(0, 0),   (real(z2), imag(z2))], line=:arrow, linecolor=:blue, linewidth=4)
	plot!([(0, 0),   (real(z3), imag(z3))], line=:arrow, linecolor=:red, linewidth=4)
	plot!([(0,-1.3), (0, 1.5)], linestyle=:dash, linecolor=:black, linewidth=1)
	plot!([z1, z3], linestyle=:dash, linecolor=:blue, linewidth=1)
	plot!([(-1.3, 0),(3, 0)], linestyle=:dash, linecolor=:black, linewidth=1)
	plot!([z2, z3], linestyle=:dash, linecolor=:blue, linewidth=1)
	plot!(circleArray, linestyle=:dot, linecolor=:blue, linewidth=1)
	annotate!(2.2, 1.05, ("z1", 12, :green))
	annotate!(1.0,-1.0,  ("z2", 12, :blue))
	annotate!(3.5, 0.12, ("z3=z1+z2", 12, :red))
end # let

# ╔═╡ d1666436-a476-4ef1-a0e9-b9ad44300a09
let
	xlim  = (-1.5, 4.5); ylim = (-1.5, 2.5)
	ϕ = atan(1/2)
	radius = 1.0
	r = sqrt(2^2 + 1^2)
	z1 = r*exp(im*ϕ)
	z2 = real(z)/r + -2*imag(z)/r*im
	z3 = z1 - z2
	circleArray = Plots.partialcircle(0, 2π, 240, radius)
	plot([(0, 0), (real(z1), imag(z1))], line=:arrow, linecolor=:green, linewidth=4, xlim=xlim, ylim=ylim, legend=false, aspect_ratio=:equal)
	plot!([(0, 0),  (real(z2), imag(z2))], line=:arrow, linecolor=:blue, linewidth=4)
	plot!([(0, 0),  (-real(z2), -imag(z2))], line=:arrow, linecolor=:lightblue, linewidth=4)
	plot!([(0, 0),   (real(z3), imag(z3))], line=:arrow, linecolor=:red, linewidth=4)
	plot!([(0,-1.3), (0, 1.5)], linestyle=:dash, linecolor=:black, linewidth=1)
	plot!([z1, z3], linestyle=:dash, linecolor=:blue, linewidth=1)
	plot!([(-1.3, 0),(3, 0)], linestyle=:dash, linecolor=:black, linewidth=1)
	plot!([-z2, z3], linestyle=:dash, linecolor=:blue, linewidth=1)
	plot!(circleArray, linestyle=:dot, linecolor=:blue, linewidth=1)
	annotate!( 2.2, 1.05, ("z1", 12, :green))
	annotate!( 1.0, -1.0, ("z2", 12, :blue))
	annotate!(-1.0, 1.1,  ("-z2", 12, :blue))
	annotate!( 2.2, 2.0,  ("z3=z1-z2=z1+(-z2)", 12, :red))
end # let

# ╔═╡ 69628aaf-2678-45ea-a142-64dd22b8b378
md"
$$x = Re(z) = 2$$
"

# ╔═╡ 6c6004db-cbca-4a03-800d-15593556baff
x = real(z)

# ╔═╡ 97ada40a-2491-4351-a061-88f6393d6ebe
md"
$$y = Im(z) = 1$$
"

# ╔═╡ 74e65c0f-470d-4a7f-ac67-174f6489b8fb
y = imag(z)

# ╔═╡ 28d542a3-6a16-4c65-a0a1-02e5d8b48439
md"
$$|z| = r = \rho = \sqrt{Re(z)^2+Im(z)^2} = \sqrt{2^2+1^2}=\sqrt{5}$$
"

# ╔═╡ 35ccdb87-274f-40e3-a40e-28e441d85655
r = abs(z)

# ╔═╡ 84f33549-7f10-451b-843f-a877a4d6c6b7
md"
$$\overline{z} = Re(z) - Im(z)$$
"

# ╔═╡ eafa2a8d-b1f4-4775-97d4-9d8f028a5324
zConj = conj(z)

# ╔═╡ 79e50aad-169b-4f7f-99d9-22d612616c06
md"
*Magnitude* or *norm* of complex vector $$z$$ is :

$$r = \rho = |z| = <z, z>^{1/2}$$ 
$$<z, z> = z^T \overline{z} = (Re(z) + Im(z)) \cdot (Re(z) - Im(z)) = x^2 + y^2$$ 
"

# ╔═╡ 3e9f674f-eb16-4f07-9a0f-65fbd4c93e54
zzConj = z * zConj

# ╔═╡ 90455237-27dc-4a4e-bbd5-c6d9931c60be
sqrt(real(z * zConj))

# ╔═╡ c03a9225-e65e-4a5f-aa44-50220d4376fa
abs(z) == sqrt(2^2 + 1^2) == sqrt(5) == sqrt(real(z * zConj))

# ╔═╡ d1b38ce9-a820-4344-9aa5-41b627f8eeb3
md"
$$\angle{z} = \phi = arctan(Im(z)/Re(z)) = arctan(1/2)$$
"

# ╔═╡ 21df5a65-3128-42b5-9f00-09dd686ced91
ϕ = angle(z)

# ╔═╡ 1fd749ff-a089-4e07-8598-80e95b964842
angle(z) == atan(imag(z)/real(z)) == atan(1/2)

# ╔═╡ e6a8e690-ea92-4657-a69e-a8d58b63b661
md"
###### Euler's formula
$$r \cdot e^{i\cdot \phi}$$
"

# ╔═╡ a5e1439e-571d-4705-895d-8ca72cd0c82c
r*exp(im*ϕ)             # (red) point on circle with radius r <> 1

# ╔═╡ 34250530-f803-44db-8f84-9828acf99d77
md"
$$e^{i\cdot \phi}$$
"

# ╔═╡ c7500900-5f03-4187-b496-d680d943454d
exp(im*ϕ)               # (blue) point on circle with radius r == 1

# ╔═╡ b28ec3ec-3a21-4ac5-a934-1573f00f5ebf
md"
$$e^{i\cdot \phi} == e^{\phi \cdot i}$$
"

# ╔═╡ ae49dcc4-4648-41a4-9bed-ef200bb66588
exp(im*ϕ) == exp(ϕ*im)  # '*' should be placed between variable name and 'im' 

# ╔═╡ 0fb878b7-1e8d-4190-a0d7-96f9bceeb606
md"
###### Euler's identity
"

# ╔═╡ 8cc23ffa-6134-4e73-88da-5fefb6ff69b8
π

# ╔═╡ 5b432ada-6370-4ff0-ad6c-2879cbb351f6
π*im

# ╔═╡ e1cf1510-9c18-43fe-ba9e-e90e2cfc6c08
exp(π*im)

# ╔═╡ 35ab6cc8-651a-4f40-89a0-deef8670cb15
abs(exp(π*im) - (-1 + 0im)) < 10.0E-10

# ╔═╡ 75c52cf4-9308-42a4-a26d-2c878fbc42d3
abs(exp(π*im) + 1 - 0im) < 10.0E-10

# ╔═╡ a5464136-3ac2-4225-b838-ec070e24b7b1
md"
###### Addition in $$\mathbb C$$
"

# ╔═╡ 21e54832-cc62-4662-94f6-7ed5399ef609
let
	z1 = Complex(2, 1)
	r1 = abs(z1)
	z2 = Complex(real(z1)/r1, -2*imag(z1)/r1)
	z3 = z1 + z2
end # let

# ╔═╡ 611e4d99-9efd-41b2-8b9c-39c76c4e7255
md"
###### Subtraction in $$\mathbb C$$
"

# ╔═╡ 8ef6ff4c-d780-4909-93f3-9fb88f89aefb
let
	z1 = Complex(2, 1)
	r1 = abs(z1)
	z2 = Complex(real(z1)/r1, -2*imag(z1)/r1)
	z3 = z1 - z2
end # let

# ╔═╡ 13fba1b1-ff87-41fe-bf30-b11ae89c072c
md"
###### Multiplication in $$\mathbb C$$
"

# ╔═╡ 21b3eb3f-023d-465e-937c-712272b5ec56
let
	z1 = Complex(2, -1)
	z2 = Complex(2, 2)
	z3 = z1 * z2
end # let

# ╔═╡ 85a7d6ba-609c-4c72-83cf-652a74f358b2
md"
---
###### *Division* in $$\mathbb C$$
[Result](https://www.intmath.com/complex-numbers/convert-polar-rectangular-interactive.php) is a complex number in *rectangular* coordinates
The formula can be found [here](https://www.cuemath.com/numbers/division-of-complex-numbers/).

$$\frac{x_1 + y_1i}{x_2 + y_2i} = \frac{x_1 + y_1i}{x_2 + y_2i} \times \frac{x_2 - y_2i}{x_2 - y_2i}=\frac{(x_1 + y_1i)(x_2 - y_2i)}{(x_2 + y_2i)(x_2 - y_2i)}$$

$$=\frac{(x_1 x_2 + y_1 y_2) + (x_2 y_1 - x_1 y_2)i}{x_2^2 + y_2^2}.$$

Example:

$$\frac{3-1i}{1+2i}=\frac{3-1i}{1+2i}\times\frac{1-2i}{1-2i}=\frac{(3-1i)(1-2i)}{(1+2i)(1-2i)}$$

$$= \frac{(3\cdot1-1\cdot2)+(1\cdot-1-3\cdot2)i}{1^2+2^2}$$
$$=\frac{1-7i}{5}=\frac{1}{5}-\frac{7}{5}i=0.2-1.4i.$$
"

# ╔═╡ 6c3a4af2-841c-4318-a4ff-9d3118bb26fd
let
	z1 = Complex(3, -1)
	z2 = Complex(1, 2)
	z3 = z1 / z2
end # let

# ╔═╡ cd2c4c81-34fc-4402-a31f-500b944b6ee6
md"
---
###### Inverse of complex number
$$z^{-1} = \frac{1}{z} = \frac{1\overline{z}}{z\overline{z}} = \frac{\overline{z}}{|z|^2}.$$

or more explicit:

$$(x+yi)^{-1} = \frac{1}{x+yi}=\frac{1}{x+yi} \times \frac{x-yi}{x-yi}=\frac{x-yi}{x^2+y^2} = \frac{x}{x^2+y^2}-\frac{yi}{x^2+y^2}.$$

Example:

$$(1+2i)^{-1}=\frac{1}{1+2i}=\frac{1+0i}{1+2i}\times\frac{1-2i}{1-2i}=\frac{1-2i}{1^2+2^2}=\frac{1-2i}{5}$$
$$=\frac{1}{5}-\frac{2}{5}i = 0.2-0.4i.$$
"

# ╔═╡ a7d52bed-d5b0-4f74-b1df-3b7acb00f8e0
let
	z1 = Complex(1, 0)
	z2 = Complex(1, 2)
	z3 = z1 / z2
end # let

# ╔═╡ c898ef69-ff89-4846-88ed-c39245c62256
md"
---
"

# ╔═╡ b0270d5f-d6b0-4a05-80f7-0b8d86ecdf84
md"
---
##### References

- **Abelson, H., Sussman, G.J. & Sussman, J.**; Structure and Interpretation of Computer Programs, Cambridge, Mass.: MIT Press, (2/e), 1996, https://sarabander.github.io/sicp/, last visit 2022/09/23

- **Feynman, R.P.**; The Feynman Lectures on Physics. Vol. I. Addison-Wesley, 1977.

- **Wikipedia**, Euler's formula, https://en.wikipedia.org/wiki/Euler%27s_formula, last visit 2022/10/09

- **Wikipedia**, Euler's identity, https://en.wikipedia.org/wiki/Euler%27s_identity, last visit 2022/10/06
"


# ╔═╡ b9e5ea98-ce2d-4594-965e-e8003a5d86ef
md"
---
##### end of ch 2.3.4
"



# ╔═╡ 0f6da3d6-bcc7-49d8-a347-c7c68b2526e9
md"
---
This is a **draft** under the Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) license. Comments, suggestions for improvement and bug reports are welcome: **claus.moebus(@)uol.de**

---
"


# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"

[compat]
Plots = "~1.35.3"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised

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deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"]
git-tree-sha1 = "c6c0f690d0cc7caddb74cef7aa847b824a16b256"
uuid = "ea2cea3b-5b76-57ae-a6ef-0a8af62496e1"
version = "5.15.3+1"

[[deps.REPL]]
deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[deps.Random]]
deps = ["SHA", "Serialization"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[deps.RecipesBase]]
deps = ["SnoopPrecompile"]
git-tree-sha1 = "612a4d76ad98e9722c8ba387614539155a59e30c"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.3.0"

[[deps.RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase", "SnoopPrecompile"]
git-tree-sha1 = "9b1c0c8e9188950e66fc28f40bfe0f8aac311fe0"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.6.7"

[[deps.Reexport]]
git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.2.2"

[[deps.RelocatableFolders]]
deps = ["SHA", "Scratch"]
git-tree-sha1 = "90bc7a7c96410424509e4263e277e43250c05691"
uuid = "05181044-ff0b-4ac5-8273-598c1e38db00"
version = "1.0.0"

[[deps.Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "838a3a4188e2ded87a4f9f184b4b0d78a1e91cb7"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.3.0"

[[deps.SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"
version = "0.7.0"

[[deps.Scratch]]
deps = ["Dates"]
git-tree-sha1 = "f94f779c94e58bf9ea243e77a37e16d9de9126bd"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.1.1"

[[deps.Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[deps.Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f"
version = "1.0.3"

[[deps.SimpleBufferStream]]
git-tree-sha1 = "874e8867b33a00e784c8a7e4b60afe9e037b74e1"
uuid = "777ac1f9-54b0-4bf8-805c-2214025038e7"
version = "1.1.0"

[[deps.SnoopPrecompile]]
git-tree-sha1 = "f604441450a3c0569830946e5b33b78c928e1a85"
uuid = "66db9d55-30c0-4569-8b51-7e840670fc0c"
version = "1.0.1"

[[deps.Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[deps.SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.0.1"

[[deps.SparseArrays]]
deps = ["LinearAlgebra", "Random"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[deps.SpecialFunctions]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "d75bda01f8c31ebb72df80a46c88b25d1c79c56d"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "2.1.7"

[[deps.Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[deps.StatsAPI]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "f9af7f195fb13589dd2e2d57fdb401717d2eb1f6"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.5.0"

[[deps.StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "d1bf48bfcc554a3761a133fe3a9bb01488e06916"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.21"

[[deps.TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"
version = "1.0.0"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
version = "1.10.1"

[[deps.TensorCore]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "1feb45f88d133a655e001435632f019a9a1bcdb6"
uuid = "62fd8b95-f654-4bbd-a8a5-9c27f68ccd50"
version = "0.1.1"

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[deps.TranscodingStreams]]
deps = ["Random", "Test"]
git-tree-sha1 = "8a75929dcd3c38611db2f8d08546decb514fcadf"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.9.9"

[[deps.URIs]]
git-tree-sha1 = "e59ecc5a41b000fa94423a578d29290c7266fc10"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.4.0"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unzip]]
git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[deps.Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "58443b63fb7e465a8a7210828c91c08b92132dff"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.14+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.12+3"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.29.0+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.1.1+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.48.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""

# ╔═╡ Cell order:
# ╟─e39aed1d-58a3-40e6-8a73-27204b742f42
# ╟─d87312c3-36cc-4d5d-9d3f-fd9da02bdadb
# ╠═90aa1e17-89f8-4884-9a3b-e58acf87bc7a
# ╟─b4ce58d8-c130-4c4f-904d-deccef8940cf
# ╟─576d32a1-4aa4-414d-84a5-99a1f5bc3208
# ╟─24a94004-d2f6-40f2-9596-0d2467fe8f4b
# ╟─c2bfc649-b5b1-4e75-83f4-76d2aa57bc5b
# ╟─eb2b46af-1eda-4b19-988d-ab8b2f7c3525
# ╟─46638e2b-704f-43df-9b61-a387762f78c9
# ╟─724ba211-8069-43b8-99b1-55d755ed3c68
# ╠═d0ed924a-378d-48d3-b0f9-ffd98ef98208
# ╟─0af14617-878d-445f-a230-fdfc04d0d5fb
# ╠═1aa0467e-da7c-4253-b2dd-8e41931337b6
# ╟─afd5a931-b8ad-46f8-b0f2-894d59d53959
# ╟─e791db57-a399-47fe-ac45-f4a61c493473
# ╟─262fb6be-d103-4407-b30b-ff5293ce2bea
# ╟─6d2eea9e-fd56-43c6-95a7-3d565dd0836a
# ╟─1e2aa2f3-81a8-4252-be4f-aa575918a67b
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