# What your favorite variation of : Euler's Formula

• end3r7
In summary, the conversation revolved around different variations of Euler's Formula, with each person sharing their personal favorite. The classic e^(ipi) + 1 = 0 was mentioned, along with other variations incorporating transcendental numbers, the imaginary unit, and the three major operations. One person even mentioned having the equation tattooed on their ankle.
end3r7
"What your favorite variation of" : Euler's Formula

I generally find that mathematicians always have a preferred way of writing an expression, whether be it because to them it's more aesthetic pleasing or easier to memorize. Few expressions, however, lend themselves to many forms as thus Euler's famous equation: $$e^{ix} = cos(x) + isin(x)$$.

I've seen it written with pi, infinite series, limits, derivatives, etc.
Here is my personal favorite
$$\sqrt{e^{-\pi}} = i^i$$

You turn. =)

I would have to stick with the classic:

e^(ipi) +1 = 0

It has multiplication, addition and exponents which are the three major operations, the multiplicative identity, the additive identity, equality, as well as two transcendental numbers and i, the imaginary unit.

I mean, after all, what more do you need that that? I think it is so beautiful I am going to have it tatooed on my ankle.

(yes, major geeky, but worth it.)

Quantumduck said:
I would have to stick with the classic:

e^(ipi) +1 = 0

It has multiplication, addition and exponents which are the three major operations, the multiplicative identity, the additive identity, equality, as well as two transcendental numbers and i, the imaginary unit.

I mean, after all, what more do you need that that? I think it is so beautiful I am going to have it tatooed on my ankle.

(yes, major geeky, but worth it.)

I also find this mathematical equation beautiful...but i wouldn't have it tatooed on me, i have poster of it on my wall

$$e^{i\pi} = 1 + i\pi - \frac{\pi^2}{2!} - \frac{i\pi^3}{3!} + \frac{\pi^4}{4!} + \frac{i\pi^5}{5!}...$$

## 1. What is Euler's Formula?

Euler's Formula is a mathematical equation that relates the trigonometric functions cosine and sine to the complex exponential function. It is written as e^(ix) = cos(x) + i*sin(x), where e is the base of the natural logarithm, i is the imaginary unit, and x is any real number.

## 2. What is the significance of Euler's Formula?

Euler's Formula is significant because it shows the deep connection between the seemingly unrelated concepts of trigonometry, complex numbers, and exponential functions. It is also used in many areas of mathematics, physics, and engineering, including signal processing, Fourier analysis, and quantum mechanics.

## 3. What is your favorite variation of Euler's Formula?

My favorite variation of Euler's Formula is the generalized version, also known as the Euler's identity, which includes the natural logarithm of a complex number. It is written as e^(ix) = cos(x) + i*sin(x) + ln(x).

## 4. How is Euler's Formula used in real-world applications?

Euler's Formula is used in many real-world applications, such as in electrical engineering for analyzing alternating current circuits, in signal processing for analyzing signals with multiple frequencies, and in physics for describing the behavior of waves and oscillations.

## 5. Can you provide an example of how Euler's Formula is used in a practical context?

Sure, one example is in digital signal processing, where Euler's Formula is used to convert a signal from the time domain to the frequency domain using the Fourier transform. This allows for the analysis of different frequencies present in the signal, which is useful in many applications such as audio and image processing.

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