Theoretical Computer Science Cheat Sheet Definitions Series ... - TUG

Wallis’ identity:
π = 2 · 2 · 2 · 4 · 4 · 6 · 6 · · ·
1 · 3 · 3 · 5 · 5 · 7 · · ·
π
Brouncker’s continued fraction expansion:
π
4 = 1 + 1 2
3
2 + 2
2+ 52
2+
2+···
72
Gregrory’s series:
π
4 = 1 − 1 3 + 1 5 − 1 7 + 1 9 − · · ·
Newton’s series:
π
6 = 1 2 + 1
2 · 3 · 2 3 + 1 · 3
2 · 4 · 5 · 2 5 + · · ·
Sharp’s series:
π
6 = √ 1 (
1 − 1
3 3 1 · 3 + 1
3 2 · 5 − 1 )
3 3 · 7 + · · ·
Euler’s series:
π 2
6 = 1 1 2 + 1 2 2 + 1 3 2 + 1 4 2 + 1 5 2 + · · ·
π 2
8 = 1 1 2 + 1 3 2 + 1 5 2 + 1 7 2 + 1 9 2 + · · ·
π 2
12 = 1 1 2 − 1 2 2 + 1 3 2 − 1 4 2 + 1 5 2 − · · ·
Partial Fractions
Let N(x) and D(x) be polynomial functions
of x. We can break down
N(x)/D(x) using partial fraction expansion.
First, if the degree of N is greater
than or equal to the degree of D, divide
N by D, obtaining
N(x)
D(x) = Q(x) + N ′ (x)
D(x) ,
where the degree of N ′ is less than that of
D. Second, factor D(x). Use the following
rules: For a non-repeated factor:
N(x)
(x − a)D(x) = A
x − a + N ′ (x)
D(x) ,
where
A =
[ ] N(x)
.
D(x)
x=a
For a repeated factor:
m−1
N(x)
(x − a) m D(x) = ∑ A k (x)
(x − a) m−k +N′ D(x) ,
where
A k = 1 k!
k=0
[ ( )]
d
k N(x)
dx k .
D(x)
x=a
The reasonable man adapts himself to the
world; the unreasonable persists in trying
to adapt the world to himself. Therefore
all progress depends on the unreasonable.
– George Bernard Shaw
Theoretical Computer Science Cheat Sheet
Derivatives:
1. d(cu)
dx
4. d(un )
dx
7. d(cu )
dx
9.
11.
13.
15.
17.
19.
21.
23.
25.
27.
29.
d(sin u)
dx
d(tan u)
dx
d(sec u)
dx
Calculus
= cdu
d(u + v)
, 2. = du
dx dx
du
= nun−1
dx ,
du
= (ln c)cu
dx
d(arcsin u)
dx
d(arctan u)
dx
d(arcsec u)
dx
d(sinh u)
dx
d(tanh u)
dx
d(sech u)
dx
d(arcsinh u)
dx
d(arctanh u)
dx
dx + dv
dx ,
d(u/v)
5. = v( du
dx
dx
d(uv)
3.
dx
= u dv
dx + v du
dx ,
) (
− u
dv
)
dx
v 2 , 6. d(ecu )
dx
, 8.
d(ln u)
dx
cu du
= ce
dx ,
= 1 du
u dx ,
= cos u du
d(cos u)
, 10. = − sin u du
dx dx
dx ,
= sec 2 u du
d(cot u)
, 12. = csc 2 u du
dx dx
dx ,
= tan u sec u du
d(csc u)
, 14. = − cot u csc u du
dx dx
dx ,
=
1 du
d(arccos u)
√ , 16. = −1 du
√
1 − u
2 dx dx 1 − u
2 dx ,
= 1 du
1 + u 2 dx
1
=
u √ du
1 − u 2 dx
, 18.
d(arccot u)
dx
, 20.
d(arccsc u)
dx
=
= −1
1 + u 2 du
dx ,
−1
u √ du
1 − u 2 dx ,
= cosh u du
d(cosh u)
, 22. = sinh u du
dx dx
dx ,
= sech 2 u du
d(coth u)
, 24. = − csch 2 u du
dx dx
dx ,
= − sech u tanh u du
d(csch u)
, 26. = − csch u coth u du
dx dx
dx ,
1 du
d(arccosh u) 1 du
= √ , 28. = √
1 + u
2 dx dx u2 − 1 dx ,
= 1 du
d(arccoth u)
1 − u 2 , 30. = 1 du
dx dx u 2 − 1 dx ,
d(arcsech u) −1
31. =
dx u √ du
d(arccsch u)
, 32. =
1 − u 2 dx dx
Integrals:
∫ ∫
∫
∫
1. cu dx = c u dx, 2. (u + v) dx =
3.
6.
8.
10.
12.
14.
−1
|u| √ du
1 + u 2 dx .
∫
u dx +
v dx,
∫
x n dx = 1
∫ ∫
1
n + 1 xn+1 , n ≠ −1, 4.
x dx = ln x, 5. e x dx = e x ,
∫
∫
dx
1 + x 2 = arctan x, 7. u dv ∫
dx dx = uv − v du
dx dx,
∫
∫
sin x dx = − cos x, 9. cos x dx = sin x,
∫
∫
∫
tan x dx = − ln | cos x|, 11.
sec x dx = ln | sec x + tan x|, 13.
arcsin x a dx = arcsin x a + √ a 2 − x 2 , a > 0,
∫
∫
cot x dx = ln | cos x|,
csc x dx = ln | csc x + cot x|,