[ Which one do you prefer working with ] Pi or e (1 Viewer)

Drongoski · Mar 28, 2009

Certainly e .. it is so rich! Just consider a few below:

$e^{x}\ \ =\ \ \sum_{k=0}^{\infty }\frac{x^{k}}{k!}\ \ =\ \ 1\ +\ \frac{x}{1!}\ +\ \frac{x^{2}}{2!}\ +\ \frac {x^{3} } {3!}\ +\ \frac{x^{4}}{4!}\ + \ \ \cdots \\ \\ \\ sin\ x\ \ =\ \ \frac{x}{1!}\ -\ \frac{x^{3}}{3!}\ +\ \frac{x^{5}}{5!}\ -\ \frac{x^{7}}{7!} \ +\ \ \ \cdots \\ \\ \\ cos\ x\ \ =\ \ \ 1\ -\ \frac{x^{2}}{2!}\ +\ \frac{x^{4}}{4!}\ - \ \frac{x^{6}}{6!}\ +\ \ \ \cdots \\ \\ \\ e^{ix}\ \ =\ \ \sum_{k=0}^{\infty }\frac{(ix)^{k}}{k!}\ \ =\ \ (1\ -\ \frac{x^{2}}{2!}\ +\ \frac{x^{4}}{4!}\ -\ \ \cdots \ )\ \ +\ \ i\ (\ \frac{x}{1!}\ -\ \frac{x^{3}}{3!}\ +\ \frac{x^{5}}{5!}\ -\ \ \cdots )\\ \\ \therefore \ \ e^{ix}\ \ =\ \ \ cos\ x\ \ \ +\ \ \ i\ sin\ x\ \ \ \ =\ \ \ cis\ x \\ \\ \\ e^{i\pi }\ =\ -1\\ \\ (\ cis\ \theta \ )^{n}\ \ =\ \ (e^{i\theta })^{n}\ \ =\ \ e^{in\theta }\ \ =\ \ cis\ n\theta \ \ =\ \ De\ Moivre's\ Theorem\ \\$

Drongoski · Mar 28, 2009

In addition:

$\\ \\ \\ e^{i\theta }\ \ \ =\ \ cos\ \theta \ \ +\ \ i\ sin\ \theta \\ \\ e^{-i\theta }\ \ =\ \ cos\ \theta \ \ -\ \ i\ sin\ \theta \\ \\ \therefore \ \ \ sin\ \theta \ \ =\ \ \frac{e^{i\theta }\ -\ e^{-i\theta }}{2\ i}\\ \\ \ \ \ \ cos\ \theta \ \ =\ \ \frac{e^{i\theta }\ +\ e^{-i\theta }}{2}\\ \\ e\ \ =\ \ 1\ +\ 1\ + \frac{1}{2!}\ +\ \frac{1}{3!}\ +\ \frac{1}{4!}\ +\ \ \cdots \\ \\ \ \lim_{x\to \infty }\ \left (1\ + \ \frac{1}{x} \right )^{x}\ \ =\ \ e$

tommykins · Mar 28, 2009

there are also series' that concern pi .

Trebla · Mar 28, 2009

$e^{i\pi}=-1$

It contains both e and pi so everyone's happy

Iruka · Mar 29, 2009

3unitz said:
*cringes at the cis*

High school folk do that.

I think its cause they don't learn Euler's formula in the HSC.

Dragonmaster262 · Mar 29, 2009

I don't even know how to put e in my calculator.

Drongoski · Mar 29, 2009

Dragonmaster262 said:
I don't even know how to put e in my calculator.

Try e¹

jessiminica · Mar 29, 2009

darkliight said:
Can't I have my pi and e it too?

That made me lol very hard

Personally, I prefer pi. Its far more elegant, and the numbers have a much nicer rhythm.

tommykins · Mar 31, 2009

Iruka said:
High school folk do that.

I think its cause they don't learn Euler's formula in the HSC.

tbh writing e^i@ is pretty annoying on paper, especially when you're doing sqrts and what not.

Dumbledore · Apr 23, 2009

pi, i can show of my memory when i'm bored without explaining what i'm reciting

Uncle · Jun 9, 2009

[maths]u(x,y)=\sum_{k=0}^{\infty }\frac{4}{\pi (2k+1)}e^{\frac{-\pi (2k+1)}{h}x}sin\frac{\pi (2k+1)}{h}y[/maths]

Last night I solved for the solution above to the partial differential equation:

[maths]\frac{\partial^2u }{\partial x^2}+\frac{\partial^2u }{\partial y^2}=0[/maths]

for
x > 0
0 < y < h
modelling the steady state two dimensional heat distribution in a slab with boundary conditions:

u = 0 when y = 0, all x > 0
u = 0 when y = h, all x > 0
u approaches 0 when x = 1, for 0 < y < h
u = 1 when x = 0, for 0 < y < h

So either way I must work with π and e.

[ Which one do you prefer working with ] Pi or e (1 Viewer)

Which one do you like pi or e

Pi

e

Drongoski

Well-Known Member

Drongoski

Well-Known Member

tommykins

i am number -e^i*pi

Trebla

Administrator

Iruka

Member

Dragonmaster262

Unorthodox top student

Drongoski

Well-Known Member

jessiminica

Member

tommykins

i am number -e^i*pi

Dumbledore

Member

Uncle

Banned

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