Eyes on e

Maurice HPM van Putten

πππeee

March 2017

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http://www.bbc.com/earth/story/20160120-the-most-beautiful-equation-is-eulers-identity2

Outline

I. Natural number e = 2.7183… (Euler)

II. Complex numbers and Euler’s identity

III. Ubiquity of e and pi

IV. Conclusions

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Boris V Balakin, Alex C Hoffmann, Pawel Kosinski & Lee D Rhyne (2010)

Terminal velocity sphere in oil and water

Terminal velocitysmooth transition in time to a constant velocity

Engineering Applications of Computational Fluid Mechanics Vol. 4, No. 1, pp. 116–126 (2010)

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V (t) =V0 1− e− t /τ( )

e = 2.718281828...

τ : time scale, set by viscosity. Relatively small for sphere dropped in oil; larger for sphere dropped in water

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Leonhard Euler (1727)

“Meditatio in Experimenta explosione tormentorum nuper instituta”

(Meditation upon experiments made recently on the firing of Canon) Swiss, 1707-1783, student of Johann

Bernouilli

6

Compound interest problem in a savings account

Initial deposit

100% interest in a one time payout

Jacob Bernoulli (1654-1705)

Final balance

7

Interest payout scheme:

once twice daily

8

Payout scheme of 100%:

once twice daily

2000W 2250W 2718W9

1000

2031

1000 2016

Account Balance over time

once twice yearly

2046 2000

1000

15002250

2017 10332018 1067

2674

1065.33

Payout scheme:

10

1000

2031

1000 2016

Account Balance over time

once twice yearly

2046 2000

1000

15002250

2017 10332018 1067

2674

1065.33

Payout scheme:

2718

1000

1033.89

1068.94

1648.68

daily

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1+ gain = final deposit

initial deposit=

27181000

= 2.718daily, n=10950 in 30 yr

1+ gain = 1+ 1

n⎛⎝⎜

⎞⎠⎟

n

= 1+ 1n

⎛⎝⎜

⎞⎠⎟

1+ 1n

⎛⎝⎜

⎞⎠⎟K 1+ 1

n⎛⎝⎜

⎞⎠⎟→ e = 2.718281828...

large n limit

Payout over n periods (“installments”)

(n times)

12

0

700

1400

2100

2800

2016 2026 2036 2046

Account balance by daily compound interest

2.71828... x 1000

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ex = 1+ x + 1

2x2 +

16x3 +L = 1+ x

1!+x2

2!+x3

3!+L

1!= 12!= 23!= 64!= 24n!= 1*2 * 3*L (n −1)*n

n! grows much faster than xn, so that the series converges for all x

n!≅ 2πn nne−n (Stirling formula)

ex = limn→∞

1+ xn

⎛⎝⎜

⎞⎠⎟

n

1+ 11!+12!+13!+14!+15!= 2.7167 e1 = 2.71828( )

1+ 21!+42!+83!+164!+325!

= 7.2667 e2 = 7.3891( )Example

Define

Can show

Definition of exponential function

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ddx

f (x) = f (x)

f (0) = 1

⎧

⎨⎪

⎩⎪

Unique solution: f (x) = ex

Alternate definition: slope(x)=exp(x)

x-axis

y-axis

tangent at x

ΔxΔy

ex

x

ex =ΔyΔx

graph

Exercise: show key property fn(x)=f(nx)

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II. Complex numbers and Euler’s identity

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Origin

positive real numbersNegative real numbers

“left” “right”

“up?”

“down?”

Leonhard Euler’s complex numbers (student of Johann Bernouilli)

(1707-1783)

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The imaginary number i

Re: real axisα =

π2

Im: imaginary axis

i2 = −1,

i = −12α = π

i

1-1

-i

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Complex numbers z

Reα

Im

1-1

z described by α, z

z with length 2 and angle 13π

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Square of a complex number

Reα

Im

2α

1-1

w = z2 described by

w = z 2

β = 2α

← z = 2, α= 13πw = z2 has length 4 and angle 2

3π

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Pythagoras on the unit circle

Im

Re axisα

x = cosα

y = sinα

z = x + iy = cosα + isinαx2 + y2 = 1cos2α + sin2α ≡ 1

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Powers on the unit circle S1

z = 1→ w = zn

w = 1, β = nα

cosα + isinα( )n = cosnα + isinnα

Im

Re axisα De Moivre

nα

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ex = 1+ x + 1

2x2 + 1

6x3 +L

Powers of the exponential function

ex( )n = enx

eix( )n = enix

eix = 1+ ix + 12(ix)2 + 1

6(ix)3 +L = 1+ ix − 1

2x2 − 1

6ix3 +L

eix = 1− 12x2 +L⎛

⎝⎜⎞⎠⎟+ i x − 1

6x3 +L⎛

⎝⎜⎞⎠⎟

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cosα + isinα( )n = cosnα + isinnα

eix( )n = enix

x =α :eiα = cosα + isinα

eiα( )n = einα = cosnα + isinnα

Powers on S1

Powers of eix

eiα = cosα + isinα :cosα =

eiα + e− iα

2= Reeiα

sinα =eiα − e− iα

2i= Im eiα

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Polar representation of complex numbers

z = r cosα + isinα( ) = reiα

zn = rneinα

cosα + isinα = cos α + 2πn( ) + isin α + 2πn( ) n∈¢( )

1= ei2πn n∈¢( )log z = log r + iα + i2πn n∈¢( )

eiα = ei α+2πn( ) n∈¢( )

N

N

N

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What have we learned

e and pi are associated by Euler’s identity

πππeee

ei0 = 1, e12π i= i, eiπ = −1, e

32π i= −i, e2π i = 1, etc

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e=2.71828..., pi=3.14159

eiπ +1= 0

Euler’s identity

Example:

“The most remarkable formula in mathematics” (Richard Feynman)

πππeee

Re

Im

α = π

i

1-1

1+ iπ

1!−π 2

2!−iπ 3

3!+L +

iπ 9

9!= −0.9760...+ 0.0069...i

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Q1. Complex numbers

Find the square z2 of the number z=1+i.

What is the square of z=1-i?

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What have we learned

Complex numbers:

a natural extension of the real numbers

exponential map w=ez in the complex plane

πππeee

Im(z)

Re(z)

w is oscillatory

w grows exponentially

w decays exponentially

w is oscillatory

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III. Ubiquity of e and pi

πππee

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Fourier series

=

+

+

+

low frequency,

large amplitude

high frequency,

small amplitude

f(x):

C1eiω0x

C2e2iω0x

C0

C3e3iω0x

(1768-1830)

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Fourier series of a periodic block function

Cn = Asinnk0ank0a

, k0 =2πL

f(x):

2a L

A

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Heat flows from hot to cold

J. Lienhard, 2013, “Entropy” - Concept Vignettes (YouTube)

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Fourier analysis of heat flow

u(t, x) :ut = uxx

u(0, x) = u0 (x)

⎧⎨⎪

⎩⎪

u = a(t)eikx :da(t)dt

= −k2a(t)

a(t) = a(0)e−k2t

https://commons.wikimedia.org/wiki/File:Heatequation_exampleB.gif

Higher modes quickly die out, leaving only k=0 at large times

exponential decay in time for k>0

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Q2. Heat flow

Consider a pair of metal bars of the same size and material, initially at T1=100 and T2=200. Upon bringing them into contact, what is their equilibrium temperature at late times?

[Hint: Heat is thermal energy, measured by temperature. Upon contact, heat flow sets in preserving total energy. The total energy at late time is the same as the total initial energy.]

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Lockyer, J.N., 1878, “Water-waves and Sound-waves”, Pop. Sc. Monthly, Vol. XIII (Appleton and Co., New York) p.166

u = Acosωt

Time harmonic motion

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P

t

u

ω =2πP

Angular frequency

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λ

x

u

k = 2πλ

Angular frequency and wave number

P

t

u

ω =2πP

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u = Acos(ωt − kx) :

utt − c2uxx = (−ω

2 + c2k2 )u = 0

Dispersion relation with velocity c

ω

k

Linear dispersion relation (dispersionless medium)

ω = ck

Wave equation

Wave propagation

time, space

39

Fourier analysis of wave motion

u(t, x) :utt = uxx u(0, x) = u0 (x)ut (0, x) = u1(x)

⎧

⎨⎪

⎩⎪

u = a(t)eikx : d 2a(t)dt 2

= −k2a(t), a(t) = a(0)e± ikt (u1 = 0)

Each Fourier mode propagates without decay: persistent wave motion

(shape may change, total energy and momentum are conserved)

Jean-Baptiste le Rond d’Alembert (1717-1983)

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Q3. Sound waves

The linear dispersion relation for sound waves shows that angular frequency and wave number have a constant ratio.

In a duet, A sings at A4 (440 Hz), B sings at E5 (660 Hz). What are the ratios the wave lengths of their sounds?

Hendrik ter Brugghen

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de Broglie matter wave

Erwin Schrodinger (1887-1991)

Louis de Broglie (1892-1987)

E = hω = ih ∂

∂tϕ

ψ (t, x) = a(x)e− iωtWave function

Probability ψ (x) 2 = a2 (x)

is eigenvalue in ih ∂∂tψ = Eψ

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Schrodinger wave function

Erwin Schrodinger (1887-1991)

ih ∂∂tψ = Hψ

H = Ek =

p2

2m→ H =

p2

2m, p = ih∂x

−ihΨ t =

h22m

Ψ xx

p = mv : Ek =12mv2 = p2

2m

43

Exponential function in relativity

light conet = x(c = 1)

x

tMinkowski spacetime

1864-1909 1879-1955

world-line

44

Hyperbolic functions

eiα = cosα + isinα

cosα =eiα + e− iα

2= Reeiα

sinα =eiα − e− iα

2i= Im eiα

cos2α + sin2α = 1

eλ = coshλ + sinhλ

coshλ = eλ + e−λ

2= cos iλ

sinhλ = eλ − e−λ

2= −isin iλ

cosh2 λ − sinh2 λ = 1

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Hyperbolic representation of a world-line

xa = (t, x), ua = dtds, dxds

⎛⎝⎜

⎞⎠⎟, ucuc = u

tut − uxux = 1

ut = coshλ, ux = sinλ

V =dxdt

=dx / dsdt / ds

=ut

ux= tanhλ

1

-1

lambda

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Rest mass energy

E = mut = mcoshλ = m 1+ sinh2 λ

pa = mua

coshλ = 1+ sinh2 λ ≅ 1+ 12sinh2 λ ≅ 1+ 1

2tanh2 λ λ <<1( )

m + 12mV 2 = E0 + Ek

E0 = mc2Restore velocity of light c:

relativistic four-momentum

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E = E02 + P2 ≅

P = mux = m dxdτ

Rest mass energy

P

E0 = mc2

relativistic x-momentum

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dispersion relation of

light

dispersion relation of m (books, your laptop, …)

E

Q4. Powering a laptop

Consider an new battery converting up to 1 g of mass into electrical power.

A laptop uses up to 100 W (Joules per second). For how many years can this battery power our laptop?

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IV. Summary

ez epitomized by Euler’s identity eipi+1=0.

ez describes exponential growth and harmonic motion, in finance, heat flow, sound waves, quantum mechanics…

ez describes our spacetime structure, of propagation of light

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