The Euler formula how was it developed?

In summary, the identity e\theta i = cos \theta + i sin \theta was discovered by Euler and can be proven using power series. The concept of writing e^{z} for complex z can also be understood through the differential equation \frac{d}{dz}e^z = e^z, e0=1. The proof of the identity involves evaluating the integral \int{\frac{\text{d}z}{z}} in two ways. This allows for the substitution of dz and z to get the identity e^{i\theta} = \cos{\theta} + i\sin{\theta}.
  • #1
Skynt
39
1
I'm reading a book called The Road to Reality by Roger Penrose, and I'm on the chapter for complex logarithms. What I don't understand is how the identity e[tex]\theta i[/tex] = cos [tex]\theta[/tex] + i sin [tex]\theta[/tex] is found through the use of complex logarithms. I also don't understand how if w = ez, z = log r + i[tex]\theta[/tex] if w is in [r, [tex]\theta[/tex]] form.

Basically, he explains in this book that e was chosen as a base in the general form w = bz because it reduces the ambiguity of bz.

I guess I'm just not seeing how the complex logarithm comes into play for the Euler formula being developed.

EDIT: Sorry about the format of the post, I was just trying latex. I hope someone can clear this up for me, because I feel it's something obvious I'm not understanding.
 
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  • #2
Off hand I don't know how Euler first discovered his identity, but I don't think natural logs were involved. However, the easiest way to prove the identity is to expand both sides into power series and see that these series are identical.
 
  • #3
First you have to think about what it means to write [itex]e^{z}[/itex] for complex z. One way is via power series as already mentioned. Another is in terms if the differential equation [itex]\frac{d}{dz}e^z = e^z[/itex], e0=1 (ez is its own derivative), which I imagine is how the concept arose historically.

If you express ei z = u(z) + iv(z) for real u and v and use the differential equation definition, then simple computation will show that ei z has length 1, its derivative has length 1, and [itex](\frac{d}{dx}e^x)(0)=i[/itex], so the curve z->ei z must trace out the unit circle in the u-v plane counterclockwise at unit-speed. (sin,cosine) also traces out the unit circle at unit speed, so Euler's formula follows.
 
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  • #4
Maybe this is the proof you mean (I saw this somewhere on mathworld I think)?
let [tex]z=\cos{\theta}+i\sin{\theta}[/tex]
then [tex]\text{d}z = -\sin{\theta} + i\cos{\theta} \, \text{d}\theta= i(\cos{\theta}+i\sin{\theta}) \, \text{d}\theta[/tex]

The integral
[tex]\int{\frac{\text{d}z}{z}}[/tex]
can be evaluated in two ways. First you can say
[tex]\int{\frac{\text{d}z}{z}} = \log{z} = \log{(\cos{\theta}+i\sin{\theta})}[/tex]
On the other hand you can substitute in dz and z to get
[tex]\int{\frac{i(\cos{\theta}+i\sin{\theta})}{\cos{\theta}+i\sin{\theta}}} \, \text{d}\theta =\int{i \, \text{d}\theta[/tex]
so
[tex]\log{(\cos{\theta}+i\sin{\theta})} = i\theta[/tex]

[tex]e^{i\theta} = \cos{\theta} + i\sin{\theta}[/tex]
 
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  • #5
I believe I have seen the page of Euler's introductio in analysin infinitorum where he gives the proof in terms of power series.
 

What is the Euler formula and why is it important?

The Euler formula, also known as Euler's identity, is a mathematical equation that relates five fundamental mathematical constants: e (Euler's number), π (pi), i (the imaginary unit), 1 (the multiplicative identity), and 0 (the additive identity). It is important because it demonstrates the deep connections between seemingly unrelated concepts in mathematics and has numerous applications in fields such as physics, engineering, and statistics.

Who developed the Euler formula?

The Euler formula was developed by Swiss mathematician Leonhard Euler in the 18th century. Euler was a prolific mathematician who made significant contributions to a wide range of fields, including calculus, number theory, and graph theory.

How was the Euler formula derived?

The Euler formula was derived by Euler using complex analysis. He used techniques from calculus and trigonometry to manipulate the exponential function eix, where i is the imaginary unit, to arrive at the elegant and concise form of the formula: eix = cos x + isin x.

What are the applications of the Euler formula?

The Euler formula has many applications in mathematics and other fields. In mathematics, it is used to prove identities, solve differential equations, and simplify complex calculations. In physics, it is used in quantum mechanics and electromagnetism. In engineering, it is used in signal processing and control theory. It also has applications in statistics, finance, and computer science.

Is the Euler formula always true?

Yes, the Euler formula is always true. It is a fundamental result in mathematics and has been rigorously proven to be true for all real numbers x. However, it is important to note that the formula is most commonly used in the context of complex numbers, where it holds true for all values of x and y in the complex plane.

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