Physical explanation of e^i*pi=-1

In summary, Euler's identity, represented by the equation e^i*pi=-1, is a fundamental relationship between the mathematical constants e, i, π, and -1. It is considered one of the most beautiful equations in mathematics and has applications in various fields, including physics and engineering. The number e represents the base of the natural logarithm and is raised to the power of i times π in this equation. The number π represents the ratio of a circle's circumference to its diameter, and is multiplied by the imaginary unit i in Euler's identity. The imaginary unit i, defined as the square root of -1, is used to represent complex numbers in mathematics and is raised to the power of π in this equation. In physics,
  • #1
THE HARLEQUIN
52
4
Hi everyone ,
i was wondering for a while to get a satisfactory proof of the equation : e^i*pi = -1
yes , i know it can be derived from euler's formula ...
which is e^i*x = cosx + isinx ( which can be proved using differential calculus )
so, e^i*pi = cos(pi) + isin(pi) ( which leads to the result we get )
but , the problem is ,i can't seem to get the physical explanation of this equation ( e.g. what e^i*pi = -1 exactly means in reality ) and the process we use to obtain it is extremely abstract( at least it seems to me ) ... is there any proof of this equation which makes a bit more sense ?

my second question is :
how do we evalute the meaning of sini and cosi ?
(suppose if i take x = i then we get ,
e^i*i = cosi + isini
=>e^-1 = cosi + isini
( if i am not wrong this equation can be solved to get individual values for sini and cosi ... but what does a complex angle mean in the first place in ? ) ... i would appreciate answers with proper physical explanation ...

thanks ALL,

THE HARLEQUIN
 
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  • #2
I'm not sure of how much of a "physical" explanation it is, but think of if as a vector in the complex plane by letting [itex]i[/itex] be the "imaginary" unit vector [itex]\vec i[/itex] and, say, [itex]\vec r[/itex] the horizontal "real" unit vector. Thus [itex]\vec v = \vec r \cos{x} + \vec i \sin{x}[/itex] represents such a vector in the complex plane. This is what [itex]\text{e}^{i x}[/itex] represents, the "real" unit vector just isn't explicitly written.

It can be noted that the unit vector [itex]\vec v[/itex] can represent any direction in the complex plane (for a real x, anyways). Changing the value of the argument merely "rotates" the vector around origin. So setting [itex]x = \pi[/itex] gives us the direction represented by the vector [itex]\vec v = - \vec r[/itex].

As for your second question, [itex]\cos z[/itex] and [itex]\sin z[/itex] for some complex [itex]z = a + i b[/itex] can be evaluated if you use their angle sum identities as well as [itex]\cos x = \frac{\text{e}^{i x} + \text{e}^{- i x}}{2}, \sin x = \frac{\text{e}^{i x} - \text{e}^{- i x}}{2 i}[/itex]. Or you can just directly use [itex]\cos z = \frac{\text{e}^{a}\text{e}^{i b} + \text{e}^{-a}\text{e}^{- i b}}{2}, \sin z = \frac{\text{e}^{a}\text{e}^{i b} - \text{e}^{-a}\text{e}^{- i b}}{2 i}[/itex].
 
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  • #3
When you use purely imaginary arguments for trig functions they become hyperbolic functions.

example cos(ix) = cosh(x).
 
  • #4
We can think of a series of small rotations starting at 1 and continuing by infinitely small shifts until having traveled the entire length of the curve connecting 1 and -1 in the complex plane. Each shift is expressed as:

##1 + i \delta##, where ##\delta## is a small angle.

The total number of shifts needed, then, is ##\frac{\pi}{\delta}##. Thus, the transformation for the whole rotation is:

##(1 + i \delta)^{\frac{\pi}{\delta}}##

Now, letting ##\delta \to 0## and replacing it by ##\frac{1}{n}## in order to use the definition:

##e = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n##,

we arrive at Euler's identity, where ##\pi## is the angle that connects 1 and -1 in the complex plane.
 
  • #5
THE HARLEQUIN said:
Hi everyone ,
i was wondering for a while to get a satisfactory proof of the equation : e^i*pi = -1

If you start out with eix (x real) and use the standard polynomial expansion for ez (1 + z/1 + z2/2!..., change z into ix and remember that i2= -1, you end up with an expansion containing both real and imaginary coefficients. Separate the real parts into one expansion and the imaginary parts into another. The real part turns out to be the familiar expansion of cos(x) and the imaginary part turns out to be the familiar expansion for sin(x) (with i as a common factor).

Based on these expansions (which, by the way , converge for all values of x), we have eix = cos(x) + i*sin(x) (for x real). Now pi is real, so ei*pi = cos(pi) + i*sin(pi) = -1 +i*0 = -1.

The second part of your question: If you change the sign of x in the formula, you get ei*(-x) = cos(-x) + i*sin(-x) = cos(x) - i*sin(x) (basic trigonometric identities).

Therefore eix + e-ix = 2*cos(x) and eix - e-ix = 2*i*sin(x) which gives you the formula cos(x) = ½(eix + e-ix). Now you want the value of cos(i). Just use the formula: cos(i) = ½(ei*i + e-i*i) =½(e-1 + e1). The formula for sin(x) will give you the value for sin(i).
 

Related to Physical explanation of e^i*pi=-1

1. What does the equation e^i*pi=-1 represent?

The equation e^i*pi=-1 is known as Euler's identity and it represents a fundamental relationship between the mathematical constants e, i, π, and -1. It is considered to be one of the most beautiful equations in mathematics.

2. How is the number e related to this equation?

The number e is a mathematical constant that represents the base of the natural logarithm. It is approximately equal to 2.71828 and is an important number in many areas of mathematics. In Euler's identity, e is raised to the power of i times π.

3. What is the significance of the number π in this equation?

The number π, also known as pi, is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is approximately equal to 3.14159 and is a fundamental constant in geometry and trigonometry. In Euler's identity, π is multiplied by the imaginary unit i.

4. What is the role of the imaginary unit i in this equation?

The imaginary unit i is defined as the square root of -1. It is used in mathematics to represent complex numbers, which have both a real and imaginary component. In Euler's identity, the imaginary unit i is raised to the power of π, resulting in a complex number.

5. How is this equation used in physics?

Euler's identity is used in physics to understand and describe the relationships between different physical quantities. It has applications in quantum mechanics, electromagnetism, and other branches of physics. It is also used in engineering and other fields to model and solve complex problems.

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