## Complex #34: sneaky use of Rouche’s theorem

**Problem #34 is: Prove that there does not exist a polynomial of the form such that for all such that .**

**Proof:** Suppose that there does exist such a polynomial. Note that for . By Rouche’s theorem, has the same number of zeros (with multiplicities) as inside the unit disk. But that is impossible, since has zeros and has at most zeros. QED.

## Complex #27: integrating over a long rectangle

**Problem 27: Show that has the same value on every straight line path (oriented in the direction) parallel to the real axis.**

Proof: Consider the rectangle , where is some given number and is a very large number. Since the integrand is entire, the integral over the boundary of is zero by the Cauchy Integral Theorem, and as , the horizontal pieces correspond to with being the line or the line , with one being negative and the other positive. Thus, the result is proved if we can show that the contributions from the vertical pieces go to zero as goes to .

Observe that the absolute value of the integrand at is , which on each vertical boundary piece is for . Integrating this over will give a value bounded by a constant times , which clearly goes to zero as .

QED.

## Complex #26: polynomial properties

**Problem #26:** If is an entire function that is not a polynomial, prove that, given arbitrary , , and integer , there exists a such that and .

The way this is worded makes my head hurt, so let’s formulate it equivalently as Not B implies Not A instead of A implies B.

**Reformulation:** Let be an entire function. Suppose that there exists a , , and an integer such that whenever . Prove that is a polynomial.

**Proof:** By the given and a limiting process, for . By the maximum modulus principle, the same fact is true for . By the Cauchy-Lagrange inequalities if , we have , where is the maximum value of for . Since this circle is completely contained in the disk , , and so we have for and thus for . By Liouville’s theorem, is a constant, and thus is a polynomial.

## Complex #30: sin(z)=z^2 solutions

One of the problems we talked about was the proof of the fact that the equation has an infinite number of complex number solutions. I thought of a couple of ways to do this. Here is one way.

Let . In rectangular coordinates, the function is

.

For a large positive integer , consider the rectangle

, where is a large constant to be chosen conveniently later. We will show that as we go counterclockwise around the boundary of , the argument of increases at least by . Then, by the argument principle, the number of zeros inside is (at least) (and there are no poles). So that is all we need for the proof, cause can be chosen to be arbitrarily large.

Increasing of the argument of :

**On the lower part of the boundary, , and

. For , this is a negative real number, so there is no change in argument as moves along the lower edge.

**On the left side of the boundary, , and

.

As moves down from (the large) to zero, the real part decreases from positive to negative, and the imaginary part also goes from positive to negative to zero. (Note: you can tell that it is still negative near zero by taking a limit — use Taylor series.) So this tells us that the argument goes from the first to the third quadrant and ends up at the negative real axis, without looping around too much. Without analyzing further, we have no idea if it goes above or below the origin, so we don’t know if the argument is increasing or decreasing. Either way, we know the argument increases or decreases by at most .

**On the right side of the boundary, , and

.

As moves up from to (the large) , the real part goes from negative to positive, and the imaginary part starts at zero, gets negative, then eventually goes positive (since increases exponentially). So in this case, we move from negative real axis to the first quadrant. Again we can conclude that the argument has changed by no more than (plus or minus).

**On the top part of the boundary, , a large number, and

.

As decreases from to , because of the largeness of , the graph looks like a small perturbation of the graph of , which is an ellipse, so we see that the goes around the origin counterclockwise (notice and switched from usual roles) times.

Thus, the total change in argument of as goes counterclockwise around the boundary of is bounded by , and since

must be an integral multiple of , we must have that the argument of makes between and revolutions counterclockwise around the origin. By the argument principle, there must be at least zeros inside .

One minor point: how large do we pick ? As long as we choose it so that is way bigger than and is way bigger than , we are fine. And we can definitely pick such a , since and increase exponentially in .

QED

## Proof of CIFD

Dear all,

Ken asked me about the proof of the Cauchy Integral Formula for Derivatives that uses the real analysis fact of differentiating under the integral sign.

Statement: If is holomorphic on a simply connected domain and is a counterclockwise oriented Jordan rectifiable curve in , then for every inside the interior of , we have

.

Proof: For fixed positive integers and in the interior of , the differential form has bounded and smooth complex coefficients on , and likewise all of its derivatives wrt are smooth and bounded. In particular, the functions mentioned are absolutely integrable over . Thus, by the theorem on differentiating under the integral sign, any partial derivative of the expression

.

may be computed by differentiating under the integral sign. For instance,

.

Therefore, by the Cauchy integral formula

,

.

QED

## Complex 3 – Cauchy Ineqs and Schwarz Lemma

Hi all – Ken U. asked me about Complex #3. I quote it here:

Suppose and are positive.

(a) Prove that the only entire functions for which for all are constant.

(b) What can you prove if for all .

**Solution:** well, there are many ways to do this. Here’s one way.

Suppose you have such an in (a). By the Cauchy Inequalities, , where is the maximum of on a disk of radius around . If , choose , and the given implies that (since is the biggest possible distance from the origin on that circle of radius around ). **Note that the inequality is true for , but by the Maximum Modulus Principle, it is also true for all such that .** The RHS of this inequality is bounded above by a constant for (see powers of ), so Liouville’s Theorem implies that is a constant. By taking a limit as , we see that that constant must be zero. Thus, , and is a constant function.

Part (b) is similar. Use the same argument, except use the Cauchy inequality for the third derivative: . Then you get using the same circles. The same argument shows that , and so is a polynomial of degree two.