Theoretical Computer Science Cheat Sheet
π Calculus
Wallis’ identity:
π =2·
2 · 2 · 4 · 4 ·6 ·6 ···
1 · 3 · 3 · 5 ·5 ·7 ···
Brouncker’s continued fraction expansion:
π
4
=1+
1
2
2+
3
2
2+
5
2
2+
7
2
2+···
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
√
3
1 −
1
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
−···
Derivatives:
1.
d(cu)
dx
= c
du
dx
, 2.
d(u + v)
dx
=
du
dx
+
dv
dx
, 3.
d(uv)
dx
= u
dv
dx
+ v
du
dx
,
4.
d(u
n
)
dx
= nu
n−1
du
dx
, 5.
d(u/v)
dx
=
v
du
dx
− u
dv
dx
v
2
, 6.
d(e
cu
)
dx
= ce
cu
du
dx
,
7.
d(c
u
)
dx
= (ln c)c
u
du
dx
, 8.
d(ln u)
dx
=
1
u
du
dx
,
9.
d(sin u)
dx
= cos u
du
dx
, 10.
d(cos u)
dx
= −sin u
du
dx
,
11.
d(tan u)
dx
= sec
2
u
du
dx
, 12.
d(cot u)
dx
= csc
2
u
du
dx
,
13.
d(sec u)
dx
= tan u sec u
du
dx
, 14.
d(csc u)
dx
= −cot u csc u
du
dx
,
15.
d(arcsin u)
dx
=
1
√
1 − u
2
du
dx
, 16.
d(arccos u)
dx
=
−1
√
1 − u
2
du
dx
,
17.
d(arctan u)
dx
=
1
1+u
2
du
dx
, 18.
d(arccot u)
dx
=
−1
1+u
2
du
dx
,
19.
d(arcsec u)
dx
=
1
u
√
1 − u
2
du
dx
, 20.
d(arccsc u)
dx
=
−1
u
√
1 − u
2
du
dx
,
21.
d(sinh u)
dx
= cosh u
du
dx
, 22.
d(cosh u)
dx
= sinh u
du
dx
,
23.
d(tanh u)
dx
= sech
2
u
du
dx
, 24.
d(coth u)
dx
= −csch
2
u
du
dx
,
25.
d(sech u)
dx
= −sech u tanh u
du
dx
, 26.
d(csch u)
dx
= −csch u coth u
du
dx
,
27.
d(arcsinh u)
dx
=
1
√
1+u
2
du
dx
, 28.
d(arccosh u)
dx
=
1
√
u
2
− 1
du
dx
,
29.
d(arctanh u)
dx
=
1
1 − u
2
du
dx
, 30.
d(arccoth u)
dx
=
1
u
2
− 1
du
dx
,
31.
d(arcsech u)
dx
=
−1
u
√
1 − u
2
du
dx
, 32.
d(arccsch u)
dx
=
−1
|u|
√
1+u
2
du
dx
.
Integrals:
1.
cu dx = c
u dx, 2.
(u + v) dx =
udx+
v dx,
3.
x
n
dx =
1
n +1
x
n+1
,n= −1, 4.
1
x
dx =lnx, 5.
e
x
dx = e
x
,
6.
dx
1+x
2
= arctan x, 7.
u
dv
dx
dx = uv −
v
du
dx
dx,
8.
sin xdx= −cos x, 9.
cos xdx= sin x,
10.
tan xdx= −ln |cos x|, 11.
cot xdx=ln|cos x|,
12.
sec xdx=ln|sec x + tan x|, 13.
csc xdx=ln|csc x + cotx|,
14.
arcsin
x
a
dx = arcsin
x
a
+
a
2
− x
2
,a>0,
Partial Fractions
Let N(x) and D(x) be polynomial func-
tions of x. We can break down
N(x)/D(x) using partial fraction expan-
sion. 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 follow-
ing 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:
N(x)
(x − a)
m
D(x)
=
m−1
k=0
A
k
(x − a)
m−k
+
N
(x)
D(x)
,
where
A
k
=
1
k!
d
k
dx
k
N(x)
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