sin(a) = | ∞ ∑ r=0 |
(-1)^{r} | a^{2r+1} |
(2r+1)! |
cos(a) = | ∞ ∑ r=0 |
(-1)^{r} | a^{2r} | |
(2r)! |
tan(a) = | sin(a) | |
cos(a) |
sin(a) = | e^{ai} - e^{-ai} |
2i |
cos(a) = | e^{ai} + e^{-ai} | |
2 |
tan(a) = | i(1-e^{2ai}) | ||
1 + e^{2ai} | where i=√-1 |
arcsin(a) = | arctan | ( | a | ) |
√(1-a^{2}) |
arcos(a) = | (π/2) - arctan | ( | a | ) | ||
√(1-a^{2}) |
arctan(a) = | ∞ ∑ r=0 |
(-1)^{r}•a^{2r+1} | where |a| ≤1 |
2r+1 |
arcsin(a) = | ∞ ∑ r=0 |
( | (2r)! | ) | a^{2r+1} | |
2^{2r}(r!)^{2} | 2r+1 | where |a| ≤ 1 which shows that it still can be a complex number |
arccos(a) = | π | - arcsin(a) |
2 |
arctan(a) = | ∞ ∑ r=0 |
(-1)^{r}a^{2r+1} | where |a| ≤ 1 a ≠ i, -i because the arctangent of i or -i does not have a solution |
2r + 1 |
arctan(a) = | a | ∞ ∑ r=0 |
r ∏ k=1 |
2ka^{2} |
1+a^{2} | (2k + 1)(1 + a^{2}) |
arctan(a) = | ∞ ∑ r=0 |
2^{2r}(r!)^{2} | a^{2r+1} | |
(2r + 1)! | (1 + a^{2})^{r+1} |
arccos(a) = -i•ln(a + i√(1-a^{2}) = | π | - arcsin(a) |
2 |
arctan(a) = | i | (ln(1-ia) - ln(1+ia)) |
2 |
sinh(a) = | e^{a} - e^{-a} |
2 |
cosh(a) = | e^{a} + e^{-a} | |
2 |
tanh(a) = | e^{2a} - 1 | |
e^{2a} + 1 |
arcsinh(a) = ln (a + √(a^{2}+1)) |
arccosh(a) = ln (a + √(a^{2}-1)) |
arctanh(a) = | ln(1+a) - ln(1-a) | = | ln((1+a)/(1-a)) | a ≠ 1, -1 |
2 | 2 |
= | ∞ ∑ r=0 |
a^{2r+1} | |a| < 1 |
2r + 1 |
sec(a) = | 1 |
cos(a) |
cosec or csc(a) = | 1 | |
sin(a) |
cot(a) = | 1 | |
tan(a) |
e^{n} := | ∞ ∑ r=0 |
n^{r} |
r! |
ln(n+1) := | ∞ ∑ r=0 |
(-1)^{r}•n^{r+1} | |
r+1 |
= | ∞ ∑ r=1 |
(-1)^{r-1} | n^{r} |
r |
ln(n) := | ∞ ∑ r=1 |
(-1)^{r-1}•(n-1)^{r} | |
r | 0 < n ≤ 2 |
ln(a+bi) := | ∞ ∑ r=1 |
(-1)^{r-1}•(a-1+bi)^{r} |
r |
^{n}C_{r} = | n! | ^{n}P_{r} = | n! | |
r!(n-r)! | (n-r)! |
cosh(a) ± sinh(a) = e | ±2a |
π | = | ∞ Π r=0 |
4r^{2} + 8r + 4 | = | ∞ Π r=0 |
(2r +2)^{2} |
2 | 4r^{2} + 8r +3 | (2r + 1)(2r+3) |
π | = | 4•tan^{-1}(1) = 4× | ∞ ∑ r=0 |
(-1)^{r} |
2r +1 |
n! ∼ √(2πn)• | ( | n | ) | n | |
e | Stirling's formula |
e | = | lim n→∞ |
(1 + n^{-1})^{n} |
Modify an equation above (which converges quickly) gives e = | ∞ ∑ r=0 |
1 |
r! |
f= | 1 | ( | 1+√5 | ) | n |
√5 | 2 |
f= | 1 | (( | 1+√5 | ) | n | - | ( | 1-√5 | ) | n | ) |
√5 | 2 | 2 |
n | ⌈ f ⌉ |
---|---|
0 | 0 |
1 | 1 |
2 | 1 |
3 | 2 |
4 | 3 |
5 | 5 |
6 | 8 |
7 | 13 |
8 | 21 |
... | ... |
Have you found an error or do you want to add more
information to these pages? You can contact me at the bottom of the home page. |