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complex analysis
De Moivre's Theorem
(
cos
x
+
i
sin
x
)
n
=
cos
(
n
x
)
+
i
sin
(
n
x
)
\small (\cos x+ i \sin x)^n=\cos(nx)+i\sin(nx)
(
cos
x
+
i
sin
x
)
n
=
cos
(
n
x
)
+
i
sin
(
n
x
)
Hyperbolic Functions
sinh
(
x
)
=
e
x
−
e
−
x
2
\sinh(x) = \frac{e^x - e^{-x}}{2}
sinh
(
x
)
=
2
e
x
−
e
−
x
Cauchy's Integral Theorem
∮
C
f
(
z
)
d
z
=
0
\oint_C f(z)dz = 0
∮
C
f
(
z
)
d
z
=
0
Euler's Identity/Formula
e
i
π
=
−
1
e^{i\pi} = -1
e
iπ
=
−
1
Trig functions in terms of e
sin
(
x
)
=
e
i
x
−
e
−
i
x
2
i
,
cos
(
x
)
=
e
i
x
+
e
−
i
x
2
\small{\sin(x) = \frac{e^{ix} - e^{-ix}}{2i}, \cos(x) = \frac{e^{ix} + e^{-ix}}{2}}
sin
(
x
)
=
2
i
e
i
x
−
e
−
i
x
,
cos
(
x
)
=
2
e
i
x
+
e
−
i
x