What is Euler formula: Definition and 14 Discussions
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:
e
i
x
=
cos
x
+
i
sin
x
,
{\displaystyle e^{ix}=\cos x+i\sin x,}
where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula.Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics".When x = π, Euler's formula evaluates to eiπ + 1 = 0, which is known as Euler's identity.
Euler gave us the below equations:
But this doesn't actually give me a number value for where the y value is when you plug in a number for x. For example, if i plug in 2pi for x, i know cosx should be 1. But that equation gives me (e^2i*pi +e^-2i*pi)/2. This doesn't give me 1. So what really...
Hello,
This is actually not homework.
I was google searching for "proving trig identities from geometric point of view), found one of the result which proves trig identities using Euler formula. I really liked it. Easier, quicker & simple.
But when the author speak about sum to product formulas...
I differentiated both sides of Euler's formula with respect to x :
e^ix = sin x + i cos x => ie^ix = cos x - i sin x
Then for comparison I multiplied both sides of Euler's formula by i:
e^ix = sin x + i cos x => ie^ix = i sin x - cos x
Each of these two procedures seems to yield the...
Hello, I am enrolled in calculus 2. Just having started a section in our textbook about integration by partial fractions, I eagerly began trying to use this integration technique wherever I could. After messing around for multiple days, I ran into this problem:
∫ 1/(x^2+1)dx
I immediately...
when I am using Euler equation for Fourier transform integrals of type \int_{-\infty}^{\infty} dx f(x) exp[ikx] I am getting following integrals:
\int_{-\infty}^{\infty} dx f(x) cos(kx) (for the real part) and
i* \int_{-\infty}^{\infty} dx f(x) sin(kx) (for its imaginary part)
I am...
Homework Statement
I have a code that successfully plots the trajectory of a ball moving under gravity and air resistance, but my method is rather long-winded and I want to use a 4d vector-first order ODE instead - but I don't know how to do it. I've tried writing some simple skeletons but...
why is psi = cos (k r - w t) + i sin ( k r - w t) = e^ [ i ( k r - w t)]?
my question precisely is why not:
1. psi = sin (k r - w t) + i cos ( k r - w t) ?
2. psi = sin (k r - w t) + i sin ( k r - w t) ?
3. psi = cos (k r - w t) + i cos ( k r - w t) ?
why not any of these three? is...
Homework Statement
Since the exercise has a graph I uploded it here :http://imageshack.us/photo/my-images/833/img9845wz.jpg/
fmax(t)=1 T=2Pi
h=1,3,5,7,...
Also I was told that I could use Eulers formula here.
Homework Equations
maybe someone could give me some tips how to make the...
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\theta i = cos \theta + i sin \theta is found through the use of complex logarithms. I also don't understand how if w = ez, z = log r +...