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|
| f
| Taylor expansion,
f(c+x) = f(c) + (f'(c)/1!).x + (f''(c)/2!).x2
+ (f'''(c)/3!).x3 + ...
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|---|
| 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 - ...
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| 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)! |
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