I will first say that I fully understand how to prove this equation from the use of power series, what I am interested in though is why $e$ and $\pi$ should be linked like they are.  As far as I know $\pi$ comes from geometry (although it does have an equivalent analytical definition), and $e$ comes from calculus.   I cannot see any reason why they should be linked and the proof doesn't really give any insights as to why the equation works.   Is there some nice way of explaining this?
2308
Could you provide a proof of Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?
47345
Just a curiosity. I just saw a video where pi was being calculated by the series  $$1-(1/3)+(1/5)-(1/7)+...$$  My question is how can $\pi$ be irrational if we can calculate it using rational numbers?
35500
I was wondering on the following and I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function itself. I can guess that it's probably not, because otherwise $e$ wouldn't be that special, but is there any proof of it?
1053
I mean, why is $F=ma$? Why not $m^{0.123}$, $a^{1.43}$ or some random non-integers or irrational?  I hope you understand that my question isn't limited just to force, energy, velocity, etc.; it also extends to the area of a square, circle, etc. and all other formulas.  I think the whole thing starts with direct proportionality.  Most of them tell about the area of a circle, $A = πr^2$, where π is 3.14159..... an irrational number! It's not about the constant. I am talking about the power of a physical quantity.  I know why it has pi. It's because we chose the constant for area square to be one. If we chose it to be 1 for the circle then the area of a square will have a constant, 1/pi.   I've edited the question to 'rational exponents' since all are giving me examples of decimal non-integers.
34735
The problem reads as follows:     Noting that $t=\frac{\pi}{5}$ satisfies $3t=\pi-2t$, find the exact value of    $$\cos(36^\circ)$$   it says that you may find useful the following identities: $$\cos^2 t+\sin^2 t = 1,\\ \sin 2t = 2\sin t\cos t,\\ \sin 3t = 3\sin t - 4\sin^3 t. $$  Do I have to do a system of linear equations in function of ..what? $t$? $\cos$?  Thanks in advance :)
43632
I've been stuck with this for a while now. I have this chain of reasoning that would imply $e^{-\pi}=e^\pi$, obviously false, since $e^\pi$ and $e^{-\pi}$ are two real distinct numbers and so I must have made an assumption somewhere that I cannot actually do. I know I have to be very careful when working with complex numbers, especially when they're in the exponents, and so I tried to make the steps as small as I could so that it would be easier to point out where it went wrong.  \begin{align} e^{-\pi}&amp;= e^{\pi\cdot -1}\tag{1}\\ &amp;=e^{\pi\cdot i^2}\tag{2}\\ &amp;= e^{\pi\cdot i\cdot i}\tag{3}\\ &amp;= \left(e^{\pi\cdot i}\right)^i\tag{4}\\ &amp;= (-1)^i\tag{5}\\ &amp;=\left(\tfrac{1}{-1}\right)^i\tag{6}\\ &amp;=\left((-1)^{-1}\right)^i\tag{7}\\ &amp;=(-1)^{-1\cdot i}\tag{8}\\ &amp;=(-1)^{-i}\tag{9}\\ &amp;=\left(e^{i\pi}\right)^{-i}\tag{10}\\ &amp;=e^{i\pi\cdot -i}\tag{11}\\ &amp;=e^{-i^2\pi}\tag{12}\\ &amp;=e^\pi\tag{13} \end{align} I suspect it has something to do with changing the base from $e$ to $-1$, but what does that mean? Are complex powers only defined for positive bases? Any help is appreciated.
65646
We all know this magical mathematical constant.   My question being , how and why pi just shows up in every other physics derivation or formula or even statistics for that matter .    Is it just coincidence or some actual reason?
72842
How do I derive:   $\ln(-1)=i\pi$ and   $\ln(-x)=\ln(x)+i\pi$  for $x&gt;0$ and  $x \in\mathbb R$  Thanks for any and all help!
82883
I always know that the natural logarithm $\log_e$ is the inverse of the exponential function $e$, but to my surprise when reading elementary functions in complex analysis, I discover that that is not always true. $$\log(e^z)=z+2k\pi i$$  My question here is that, why is the above expression true?
105813
So in my textbook, it states that the $E=hf$ equation applies to electrons, and all particles, not just photons.   But in order to prove this, wouldn't these particles need to have zero mass, to satisfy $E=pc$ from Einstein's equation?  In Einstein's theory of relativity $E^2= p^2c^2 + m^2c^4$, we get $E=pc$ because we consider the mass to be zero.   For $E=hf$ to work for other particles, don't we need to assume electrons have zero rest mass?
119522
I understand the proof of this first formula.  I am confused about where he got the first E[min(X, k)] formula. It seems to have just come out of nowhere. And also, can someone explain why the two formulas are correct please?
97123
I'm very eager to know and understand the definition of $e$. Textbooks define $e$ as follows $$ e = \lim_{p\to\infty} \left[1+\frac1{p}\right]^p \approx 2.71828 $$ Is there an &quot;easy to understand&quot; proof of this? I'm really looking for a derivation of this which is very intuitive and easy to comprehend. By the way I'm watching .
19734
I was playing around with Euler's identity   $$\begin{align}e^{i\pi}&amp;=-1\\\\\frac{\log{(-1)}}{i}&amp;=\pi\\\\\frac{x \log(-1)}{i}&amp;=\pi x \end{align}$$   And when I had $x=2$  $$\frac{\log(1)}{i}=2\pi$$Which is equivalent to $$\frac{0}{i}=2\pi$$  How can this be?
99574
How do I go about proving this? I know one method is:  $\eqalign{    \cos (90^\circ + \theta ) &amp;\equiv \cos90^\circ \cos\theta  - \sin90^\circ \sin\theta   \cr    &amp;  \equiv (0)(\cos\theta ) - (1)(\sin \theta )  \cr    &amp;  \equiv  - \sin \theta  \cr} $  I'd appreciate others, particularly ones that will allow me to visualize this identity on the unit circle.  Thanks alot.
11871
I've heard a lot about this number $e$. Why is it so important? How does it fit into the 'bigger picture' of mathematics? How is it calculated and used?
41446
A proof of the inequality using properties of $\pi$ and $e$, for example, is what I'm looking for. Not calculator approximations showing the inequality holds.
92794
For a complex number $A$ and a real number $B$, when does the well-known formula $(e^A)^B = e^{AB}$ fail? Or does it hold at all for complex A?     Since $e^{2\pi i} = 1$, if this formula holds for all complex numbers $A$ and real numbers $B$, then it would imply that $e^{2\pi ti}=( e^{2\pi i} )^t = 1^t=1$ for all $t$, which is obviously false.