×
First Delver! (I was the first non-tester/dev
to conquer TCB.)
d/dy

×
First Delver! (I was the first non-tester/dev
to conquer TCB.)
d/dy

×
To proove this we need to use Taylorseries. The point is that every funtion is also writeable as a polynome (aka Taylorserie). (That’s not completely true, but good enough to prove : e^(pi.i)+1=0
f(x) = a0 + a1x + a2x2 + a3x3 + a4x4 + ... + anxn + ...
For most funtions it’s not eusy to calculate, but for us it’s interesting enough. Because we only need functions: e^x and cos(x) and sin(x)..
We start with ex. We know, when we derive f(x)=ex it’s the same, f'(x)=ex. So we’re gonna use this to derive the following function
f(x) = ex = a0 + a1x + a2x2 + a3x3 + a4x4 + ...
f(0) = e0 = 1 = a0 + a10 + a202 + a303 + a404 + ... = a0 so a0 = 1
f'(x) = ex=a1 + 2*a2x + 3*a3x2 + 4*a4x3 + ... because (xn)' = nxn-1
f'(0) = e0 = 1 = a1 + 2*a20 + 3*a303 + 4*a403 + ... = a1 so a1 = 1
f"(x) = ex = 2*a2 + 2*3*a3x + 3*4*a4x2 + ...
f"(0) = e0 = 1 = 2*a2 + 2*3*a3*0 + 3*4*a4*02 + ... = 2*a2 so a2 = 1/2
f"'(x) = ex = 2*3*a3 + 2*3*4*a4x + ...
f"'(0) = e0 = 1 = 2*3*a3 + 2*3*4*a40 + ... = (2*3)a3 so a3 = 1/(2*3)
f""(x) = ex = 2*3*4*a4 + ...
f""(0) = e0 = 1 = 2*3*4*a4 + ... = 2*3*4*a4 so a4 = 1/(2*3*4)
So:
a0 = 1/1, a1 = 1/1, a2 = 1*2, a3 = 1/(1*2*3), a4 = 1/(1*2*3*4)
This can be shortened n! = 1*2*3*...*n, 0! = 1 (faculty), so we get an = 1/n!
ex = 1 + x + (1/2!)x2 + (1/3!)x3 + (1/4!)x4 + ... + (1/n!)xn + ...
We can olso do this for sin(x) en cos(x). sin'(x) = cos(x), cos'(x) = -sin(x), sin(0)=0, cos(0)=1, That’s all we need to know for Taylorseries.
f(x) = sin(x) = a0 + a1x + a2x2 + a3x3 + a4x4 + ...
f(0) = sin(0) = 0 = a0 + a10 + a202 + a303 + a404 + ... = a0 so a0 = 0
f'(x) = cos(x) = a1 + 2*a2x + 3*a3x2 + 4*a4x3 + ... (xn)' = nxn-1
f'(0) = cos(0) = 1 = a1 + 2*a20 + 3*a303 + 4*a403 + ... = a1 so a1 = 1
f"(x) = -sin(x) = 2*a2 + 2*3*a3x + 3*4*a4x2 + ...
f"(0) = -sin(0) = 1 = 2*a2 + 2*3*a3*0 + 3*4*a4*02 + ... = 2*a2 so a2 = 0
f"'(x) = -cos(x) = 2*3*a3 + 2*3*4*a4x + ...
f"'(0) = -cos(0) = -1 = 2*3*a3 + 2*3*4*a40 + ... = (2*3)a3 so a3 = -1/(2*3)
f""(x) = sin(x) = 2*3*4*a4 + ...
f""(0) = sin(0) = 0 = 2*3*4*a4 + ... = 2*3*4*a4 so a4 = 0 We can shorten it the kinda the same as above, what gives us: an = 0 als n = even, and an = (-1)(n-1)/2/n! if n = odd.
sin(x) = x - (1/3!)x3 + (1/5!)x5 - (1/7!)x7 + ...
You can do this with cos(x) as wel, but you can try that yourself, I’ll give you the solution:
cos(x) = 1 + (1/2!)x2 + (1/4!)x4 + (1/6!)x6 + ...
We now have:
ex = 1 + x + (1/2!)x2 + (1/3!)x3 + (1/4!)x4 + ...
sin(x) = x - (1/3!)x3 + (1/5!)x5 - (1/7!)x7 + ...
cos(x) = 1 - (1/2!)x2 + (1/4!)x4 - (1/6!)x6 + ...
You see, it all looks alike, if we add Complex numbers, it’s gonna look even more alike.
i
We now insert the number i. (i2 = -1) We’re gonna use this toget negative numbers. We’re going to put “ix” in ex, so we get:
eix = 1 + ix + (1/2!)(ix)2 + (1/3!)(ix)3 + (1/4!)(ix)4 + ...
To write it easier, we’re gonna check a table
i = i
i2 = -1
i3 = i2 * 1 = -1 * i = -i
i4 = i2 * i2 = (-1) * (-1) = 1
i5 = i4 * i = 1 * i = i
If we fill this in we get:
eix = 1 + ix - (1/2!)x2 - i(1/3!)x3 + (1/4!)x4 + ...
This looks pretty, try to write cos(x) + i * sin(x) :
cos(x) + i * sin(x) = 1 + ix - (1/2!)x2 - i(1/3!)x3 + (1/4!)x4 + ... = eix !!!!!!
And that’s what it was all about
eix = cos(x) + i * sin(x) !
And now it’s simple
ei*pi = cos(pi) + i * sin(pi) = -1 + i * 0
ei*pi = -1 or ei*pi + 1 = 0

×DROD number: 688
20th Skywatcher!

If you want to play KDD AE in DROD 5.0

If you want Neather sprites

DROD Statistics: (rooms/secrets/challenges)

Official Holds Total Room Count (without KDD AE): 4197
Official Holds Total Secret Room Count: 357
Official Holds Challenges: 344

Nice links:
FlashDROD
Levelmaps
Challenges
Hall of Legendary Smitemasters
Idea Generator
Room Search
Architects Edition

×DROD number: 688
20th Skywatcher!

If you want to play KDD AE in DROD 5.0

If you want Neather sprites

DROD Statistics: (rooms/secrets/challenges)

Official Holds Total Room Count (without KDD AE): 4197
Official Holds Total Secret Room Count: 357
Official Holds Challenges: 344

Nice links:
FlashDROD
Levelmaps
Challenges
Hall of Legendary Smitemasters
Idea Generator
Room Search
Architects Edition

×
First Delver! (I was the first non-tester/dev
to conquer TCB.)
d/dy

×DROD number: 688
20th Skywatcher!

If you want to play KDD AE in DROD 5.0

If you want Neather sprites

DROD Statistics: (rooms/secrets/challenges)

Official Holds Total Room Count (without KDD AE): 4197
Official Holds Total Secret Room Count: 357
Official Holds Challenges: 344

Nice links:
FlashDROD
Levelmaps
Challenges
Hall of Legendary Smitemasters
Idea Generator
Room Search
Architects Edition