# Complex Analysis: #7 Some Standard Theorems of Complex Analysis

**Corollary (Goursat’s Theorem)**

The derivative of every analytic function is again analytic. Thus every analytic function has arbitrarily many continuous derivatives.

We can also complete the statement of theorem 3

**Theorem 10 (Morera’s Theorem)**Let G ⊂ ℂ be a region and let f : G → ℂ be continuous such that ∫

_{γ}f(z)dz = 0, for all closed paths which are the boundaries of triangles completely contained within G. Then f is analytic.

*Proof*

According to theorem 3, there exists an antiderivative F : G → ℂ, with F' = f. Thus, by Goursat’s Theorem, f is also analytic.

**Theorem 11 (Cauchy’s estimate for the Taylor coefficients)**Again, let f : G → ℂ be analytic, z

_{0}∈ G, r > 0 is such that D(z

_{0}, r) = {z : |z − z

_{0}| ≤ r} ⊂ G, and

for all z ∈ D(z

_{0}, r). Since f is continuous and D(z

_{0}, r) is compact, we must have |f| being bounded in D(z

_{0}, r). Let M > 0 be such that |f(z)| ≤ M for all z ∈ D(z

_{0}, r). Then we have

**Deﬁnition 5**

Let the function f : ℂ → ℂ be defined throughout the whole complex plane, and let it be analytic everywhere. Then we say that f is an

*entire function*.

**Theorem 12**A bounded entire function is constant.

*Proof*

Assume that the entire function f : ℂ → ℂ is bounded with |f(z)| ≤ M say, for all z ∈ ℂ, where M > 0 is ﬁxed. Thus |c

_{n}| ≤ M/r

^{n}= 0 for all r > 0. This can only be true if c

_{n}= 0 for all n > 0.

**Deﬁnition 6**

A ﬁeld is called

*algebraically closed*if every polynomial within the ﬁeld of degree greater than or equal to one has a root.

**Theorem 13 (The Fundamental Theorem of Algebra)**ℂ is algebraically closed.

*Proof*

Let f(z) = ∑a

_{k}z

^{k}, (with k = 0, . . .,n), with n ≥ 1 and a

_{n}≠ 0 be a polynomial of degree n. Looking for a contradiction, we assume that there is

*no root*, that is, f(z) ≠ 0 for all z ∈ ℂ.

For z ≠ 0, we have

Note that for a and b arbitrary numbers, we have |a| = |a + b − b| ≤ |a + b| + |b| or |a + b| ≥ |a| − |b|, and more generally, |a + b

_{1}+ · · · + b

_{n}| ≥ |a| − |b

_{1}| − · · · − |b

_{n}|.

Since |a

_{n}|/2 remains constant, |z

^{n}| · |a

_{n}|/2 becomes arbitrarily large, as |z| → ∞. Therefore |f(z)| → ∞ when |z| → ∞. That is to say, if M > 0 is given, then there exists an r > 0 such that |f(z)| > M for all z with |z| > r. That is, |1/f(z)| < 1/M for |z| > r. Now, since f(z) ≠ 0 always, and f (being a polynomial) is an entire function, we have that 1/f is also an entire function. It is bounded outside the closed disc D(0, r), but since the function is continuous, and D(0, r) is compact, it is also bounded on D(0, r). Thus it is bounded throughout ℂ, and is therefore constant, by theorem 12. Therefore, the polynomial f itself is a constant function. This contradicts the assumption that f is of degree greater than zero.

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