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CHEAT SHEET
Definition 1. If z = x + iy then ez^ is defined to be the complex number ex(cos y + i sin y).
Proposition 2 (De Moivre’s Formula). If n = 1, 2 , 3 ,... then
(cos θ + i sin θ)n^ = cos nθ + i sin nθ.
Definition 3. A domain is an open connected set.
Definition 4. A complex-valued function f is said to be analytic on an open set D if it has a derivative at every point of D.
Theorem 5. If f (z) = u(x, y) + iv(x, y) is analytic in D then the Cauchy-Riemann equations
∂u ∂x
∂v ∂y
∂u ∂y
∂v ∂x
must hold at every point in D.
Theorem 6 (The Fundamental Theorem of Algebra). Every nonconstant polynomial with complex coefficients has at least one zero in C.
Theorem 7. Given any complex number z, we define
sin z =
eiz^ − e−iz 2 i
, cos z =
eiz^ + e−iz 2
Definition 8. The principal value of the logarithm is defined as
Log z := Log |z| + i Arg z.
Theorem 9 (The M L-inequality). If f is continuous on the contour Γ and if |f (z)| ≤ M for all z on Γ then (^) ∣ ∣ ∣ ∣
Γ
f (z) dz
∣ ≤^ M^ ·^ length(Γ).
Theorem 10 (Independence of Path). Suppose that the function f (z) is continuous in a domain D and has an antiderivative F throughout D. Then for any contour Γ in D with initial point zI and terminal point zT we have (^) ∫
Γ
f (z) dz = F (zT ) − F (zI ).
Definition 11. A domain D is simply connected if one of the following (equivalent) conditions holds:
- every loop in D can be continuously deformed in D to a point.
- if Γ is any simple closed contour in D then int(Γ) ⊆ D.
Theorem 12 (Cauchy’s Integral Theorem). If f is analytic in a simply connected domain D then ∫
Γ
f = 0 for all loops Γ ⊆ D.
Theorem 13 (Morera’s Theorem). If f is continuous in a domain D and if ∫
Γ
f = 0 for all loops Γ ⊆ D
then f is analytic in D.
2
Theorem 14 (Cauchy’s (generalized) Integral Formula). If f is analytic inside and on the simple positively oriented loop Γ and if z is any point inside Γ then
f (n)(z) =
n! 2 πi
Γ
f (ζ) (ζ − z)n+^
dζ
for n = 1, 2 , 3 ,.. ..
Theorem 15 (Liouville’s Theorem). The only bounded entire functions are the constant functions.
Theorem 16 (The Maximum Modulus Principle). A function analytic in a bounded domain and continuous up to and including its boundary attains its maximum modulus on the boundary.
Theorem 17 (The Taylor Series Theorem). If f is analytic in the disk |z − z 0 | < R then the Taylor series (^) ∞ ∑
n=
f (n)(z 0 ) n!
(z − z 0 )n
converges to f (z) for all z in the disk. Furthermore, the convergence is uniform in any closed subdisk |z − z 0 | ≤ R′^ < R.
Definition 18. A point z 0 is called a zero of order m for f if f is analytic at z 0 and if f (n)(z 0 ) = 0 for 0 ≤ n ≤ m − 1 but f (m)(z 0 ) 6 = 0.
Definition 19. If f has an isolated singularity at z 0 and if
f (z) =
∑^ ∞
n=−∞
an(z − z 0 )n
is its Laurent expansion around z 0 then
(1) If an = 0 for all n < 0 then f has a removable singularity at z 0. (2) If a−m 6 = 0 for some m > 0 but an = 0 for all n < −m then f has a pole of order m at z 0. (3) If an 6 = 0 for infinitely many negative n then f has an essential singularity at z 0.
Theorem 20 (Picard’s Theorem). A function with an essential singularity assumes every complex number, with possibly one exception, as a value in any neighborhood of this singularity.
Definition 21. If f has an isolated singularity at z 0 then the coefficient a− 1 of 1/(z − z 0 ) in the Laurent expansion of f around z 0 is called the residue of f at z 0.
Theorem 22 (The Residue Theorem). If Γ is a simple positively oriented loop and f is analytic inside and on Γ except at the points z 1 ,... , zn inside Γ then
1 2 πi
Γ
f (z) dz =
∑^ n
k=
Res(f ; zk).
Theorem 23 (The Argument Principle). If f is analytic and nonzero at each point of a simple positively oriented loop Γ and is meromorphic inside Γ then
1 2 πi
Γ
f ′(z) f (z)
dz = Z − P,
where Z is the number of zeros of f inside Γ and P is the number of poles inside Γ, counted with multiplicity.
Theorem 24 (Rouch´e’s Theorem). If f and g are analytic inside and on a simple loop Γ and if
|f (z) − g(z)| < |f (z)| for all z on Γ
then f and g have the same number of zeros (counting multiplicities) inside Γ.
