Calculus
| f | Taylor expansion, f(c+x) = f(c) + (f'(c)/1!).x + (f''(c)/2!).x2 + (f'''(c)/3!).x3 + ... |
|---|---|
| 1 / (1+x)r | 1 - r x + r.(r+1)/2! x2 - r.(r+1).(r+2)/3! x3 + ... |
| 1 / (1 - x)r | 1 + r x + r.(r+1)/2! x2 + ... |
| log(1+x) | x - 1/2 x2 + 1/3 x3 - ... |
| log(1 - x) | - x - 1/2 x2 - 1/3 x3 - ... |
| sin x | x - x3/3! + x5/5! - x7/7! + ... |
| cos x | 1 - x2/2! + x4/4! - x6/6! + ... |
| sinh x | 1 + x3/3! + x5/5! + ... |
| cosh x | 1 + x2/2! + x4/4! + ... |
| f(x) | derivative, d/dx f(x) |
|---|---|
| c | 0 |
| xn | n.xn-1 |
| loge x | 1/x |
| x-n | -n.x-n-1 |
| ex | ex |
| xx | xx(ln(x) + 1) |
| sin x | cos x |
| cos x | - sin x |
| tan x | 1/cos2x = 1+tan2x |
| sinh x | cosh x |
| cosh x | sinh x |
| tanh x | 1 - tanh2 x = 1/cosh2 x |
| f(x)+g(x) | f' + g', where f'=d/dx f, & g'=d/dx g |
| f(x) g(x) | f'g + fg' |
| f(x)/g(x) | (f'g - fg')/g2 |
| f(g(x)) | f'(g(x)) g'(x), the chain rule |
| f -1x | 1/(f'(f -1x)), if f -1 is the inverse function of f, x=f y, f -1(f(y))=y, f(f -1x)=x |
| f(x) | indefinite integral ∫ f(x) dx |
|---|---|
| xn | (1/(n+1)).xn+1, n≠-1 |
| x-1 | log x |
| f(x) | indefinite ∫ f (x) dx | -∞∫+∞ f(x) dx |
|---|---|---|
| 1 / (a+x2)k | x .{hg([1/2,k], [3/2], -x2/a)} / ak | √π.Γ(k - 1/2) / {a(k-1/2).Γk} |
| x2 / (a+x2)k | ? | √π.Γ(k - 3/2) / {2.a(k-3/2).Γk} |
| meaning | |
|---|---|
| hg() | hypergeometric() |
| Γ | Γ function; for int n, Γn=(n-1)! |