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Fermi's Piano Tuner Problem

How Old is Old?

If the Terrestrial Poles were to Melt...

Sunlight Exerts Pressure

Falling Eastward

What if an Asteroid Hit the Earth

Using a Jeep to Estimate the Energy in Gasoline

How do Police Radars really work?

How "Fast" is the Speed of Light?

How Long is a Light Year?

How Big is a Trillion?

"Seeing" the Earth, Moon, and Sun to Scale

Of Stars and Drops of Water

If I Were to Build a Model of the Cosmos...

A Number Trick

Designing a High Altitude Balloon

Pressure in the Vicinity of a Lunar Astronaut Space Suit due to Outgassing of Coolant Water

Calendar Calculations

Telling Time by the Stars - Sidereal Time

Fields, an Heuristic Approach

The Irrationality of

The Irrationality of

The Number (i)i

Estimating the Temperature of a Flat Plate in Low Earth Orbit

Proving that (p)1/n is Irrational when p is a Prime and n>1

The Transcendentality of

Ideal Gases under Constant Volume, Constant Pressure, Constant Temperature and Adiabatic Conditions

Maxwell's Equations: The Vector and Scalar Potentials

A Possible Scalar Term Describing Energy Density in the Gravitational Field

A Proposed Relativistic, Thermodynamic Four-Vector

Motivational Argument for the Expression-eix=cosx+isinx

Another Motivational Argument for the Expression-eix=cosx+isinx
Calculating the Energy from Sunlight over a 12 hour period
Calculating the Energy from Sunlight over actual full day
Perfect Numbers-A Case Study
Gravitation Inside a Uniform Hollow Sphere
Further note on Gravitation Inside a Uniform Hollow Sphere
Pythagorean Triples
Black Holes and Point Set Topology
Additional Notes on Black Holes and Point Set Topology
Field Equations and Equations of Motion (General Relativity)
The observer in modern physics
A Note on the Centrifugal and Coriolis Accelerations as Pseudo Accelerations - PDF File
On Expansion of the Universe - PDF File
Another Motivational Argument for the Expression: eiq= cos q+ i sin q

Problem: Show that eiq = cos q+ i sin q, where i = square root of negative one.

Solution: Let z = x + iy be any complex number. We know, from geometry, that

z = x + iy = r(cos q + i sin q).

In the previous article on eix, we used the theory of differential equations to establish the required identity. This time, we will use the natural logarithm function ln(z) to establish that same identity.
Let us form the function ln(z):

ln(z) = ln(x + iy)
= ln [r(cos q + i sin q)]
= ln(rho) + ln(cos q + i sin q).

The first term in the third line, ln(rho), involves the real number rho, and so will concern us no further here. The second term, ln(cos q + i sin q), involves a complex number whose magnitude is unity.
Let us set

u(q) = ln(cos q + i sin q)


eu(q) = eln(cos q + i sin q) = cos q + i sin q.

Our problem thus reduces to showing that u(q) = iq .We notice immediately that

eu(0) = cos 0 + i sin 0 = 1

which gives us the identity

u(0) = 1.

We now differentiate the expression eu = cos q + i sin q to obtain

deu = eu du = (- sin q + i cos q) dq
eu = (- sin q + i cos q) dq/du = cos q + i sin q.

We now have the derivative dq/du:

dq/du = (cos q + i sin q)/(- sin q + i cos q)
du = [(- sin q + i cos q)/(cos q + i sin q)] dq

Multiplying by unity in the form (cos q - i sin q)/(cos q - i sin q) allows us to simplify the right-hand side, giving

du = i dq.

Integrating, we acquire

u = iq + C.

But since we already know that u = 0 when q = 0, we have that the constant C = 0. Therefore

u = iq,

which establishes the required identity.

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