Minimum Modulus Principle and Maximizing |sin(z)|

Simplicity is the ultimate sophistication, Anonymous

The Minimum Modulus Principle

The Minimum Modulus Principle. Let $f$ be a function that is analytic and non-zero in a region (open, connected set) $A$. Then, $|f(z)|$ cannot have a local minimum in $A$ unless $f$ is constant.

The “Landscape” Analogy

To visualize this, imagine the graph of $|f(z)|$ as a three-dimensional landscape over the complex plane:

• The Maximum Modulus Principle tells us that for a non-constant analytic function, “mountain peaks” (strict local maxima) can never exist inside the region; they are always pushed to the boundary.
• The Minimum Modulus Principle addresses “pits” or “valleys.” If the function never touches zero (our “sea level”), then there are no no interior “valleys” or “pits” either. The pits are also forced to the boundary.
• Crucial Note: If $f(z)$ does equal zero at some point $z_0$ inside $A$, then $z_0$ is a trivial local minimum (since $|f(z)| \ge 0$, so $\exist z_0, |f(z)| \ge |f(z_0)| = 0$, which is clearly the smallest possible value). This is why the “non-zero” condition is required for the theorem to be non-trivial.

Proof.

1. Assume $f(z)$ is analytic and $f(z) \neq 0$ for all $z \in A$.
2. Define the reciprocal function $g(z) = \frac{1}{f(z)}$.
3. Since $f$ is analytic and its denominator is never zero in $A$, $g$ is well-defined and analytic in $A$.
4. By the Maximum Modulus Principle, $|g(z)|$ cannot have a local maximum in $A$ unless $g$ is constant.
5. Observe inverse relationship between the magnitude of a number and the magnitude of its reciprocal: a local maximum for $|g(z)| = \left| \frac{1}{f(z)} \right| = \frac{1}{|f(z)|}$ corresponds exactly to a local minimum for $|f(z)|$.

   Maximum modulus theorem. A non-constant analytic function cannot have a strict local maximum inside its domain.

6. Therefore, $|f(z)|$ cannot have a local minimum in $A$ unless $f$ is constant. $\square$

Application: Maximizing $|\sin(z)|$

Problem: Find the maximum value of $|\sin(z)|$ on the square domain $D = \{ z = x + iy \mid 0 \le x \le 2\pi, 0 \le y \le 2\pi \}$.

Solution.

1. Expanding $\sin(z)$.

To find the modulus, we first express $\sin(z)$ in terms of real and imaginary parts. Let $z = x + iy$:

$$ \begin{aligned} sin(z) = \sin(x+iy) &=[\text{Addition formula for sine}] \sin(x)\cos(iy) + \cos(x)\sin(iy) \\[2pt] &\text{Recall the hyperbolic identities: } \cos(iy) = \cosh(y), \sin(iy) = i\sinh(y)\\[2pt] &= \sin(x)\cosh(y) + i\cos(x)\sinh(y). \end{aligned} $$

2. Computing the Modulus Squared.

We want to maximize $|\sin(z)|$. It is mathematically easier to maximize the square, $|\sin(z)|^2$, because the square root function is monotonic (if $A > B$, then $\sqrt{A} > \sqrt{B}$). Using the formula $|u+iv|^2 = u^2 + v^2$:

$$ \begin{aligned} |\sin(z)|^2 = [\sin(x)\cosh(y)]^2 + [\cos(x)\sinh(y)]^2 &= \sin^2(x)\cosh^2(y) + \cos^2(x)\sinh^2(y) \\[2pt] &\text{Use the identity: } \cosh^2(y) = 1 + \sinh^2(y)\\[2pt] &= \sin^2(x)(1 + \sinh^2(y)) + \cos^2(x)\sinh^2(y) \\[2pt] &\text{Factor out } \sinh^2(y) \\[2pt] &= \sin^2(x) + \sinh^2(y)(\underbrace{\sin^2(x) + \cos^2(x)}_{1}). \end{aligned} $$

Therefore, $\boxed{|\sin(z)|^2 = \sin^2(x) + \sinh^2(y)}$

3. Applying the Maximum Modulus Principle

Since $\sin(z)$ is an entire function (analytic everywhere), its maximum modulus on the closed square $D$ must occur on the boundary. We seek to maximize the sum of two non-negative terms: $\sin^2(x)$ and $\sinh^2(y)$.

$\sin(z)$ is one of the classical entire functions, along with $e^z, \cos(z)$, polynomials, etc. $\sin z=\sin x\cosh y+i\cos x\sinh y,u(x,y)=\sin x\cosh y, v(x,y)=\cos x\sinh y.$ Its partial derivatives ($u_x=\cos x\cosh y, v_y=\cos x\cosh y, u_y=\sin x\sinh y, v_x=-\sin x\sinh y$) satisfy the Cauchy–Riemann equations ($u_x = v_y, u_y=-v_x$).

Since u and v have continuous partial derivatives everywhere (their partial derivatives $u_x, u_y, v_x, v_y$ are compositions of continuous functions (sin, cos, sinh, cosh) and are therefore continuous everywhere), $\sin(z)$ is analytic everywhere (by the sufficiency of the Cauchy-Riemann equations).

By the Maximum Modulus Principle, the maximum of a non-constant analytic function on a compact set (like this closed square) must occur on the boundary.

We need to maximize the sum of two positive terms: $\sin^2(x) + \sinh^2(y)$.

1. The function $\sinh(y)$ is strictly increasing on the entire real line. On our domain $y \in [0, 2\pi]$, the maximum value occurs at the largest $y, y = 2\pi$.

   Recall: $\sinh y=\frac{e^y-e^{-y}}{2}, \frac{d}{dy}\sinh y=\frac{e^y-(-e^{-y})}{2} = \frac{e^y+e^{-y}}{2} \gt 0$

2. The function $\sin^2(x)$ oscillates between 0 and 1. The maximum value is 1, it occurs when $\sin(x) = \pm 1$. On our domain $x \in [0, 2\pi]$, the points where sine is $\pm 1$ are: $x = \frac{\pi}{2} \text{ and } x = \frac{3\pi}{2}$

4. Conclusion

$0 \le sin^2(x) \le 1$ & $sin^2(x) = 1$ where $x = \frac{\pi}{2} \text{ or } \frac{2\pi}{2}$

To get the global maximum, we need both terms to be at their peak simultaneously, $z_1 = \frac{\pi}{2} + i2\pi, z_2 = \frac{3\pi}{2} + i2\pi$.

Hence, the maximum value of $|\sin(z)|$ occurs at the points $\frac{\pi}{2} + 2\pi i$ and $\frac{3\pi}{2} + 2\pi i$.

The maximum value of the modulus squared is: $|\sin(z)|^2_{max} = \sin^2\left(\frac{\pi}{2}\right) + \sinh^2(2\pi) = 1 + \sinh^2(2\pi)$

Using the identity $1 + \sinh^2(\theta) = \cosh^2(\theta)$, the maximum modulus is: $\max_{z \in D} |\sin(z)| = \sqrt{\cosh^2(2\pi)} = \cosh(2\pi)$

Why use the Maximum Modulus Principle here? Even though we found the maximum by looking at $x$ and $y$ separately, the principle guarantees that we don’t need to check the interior of the square (the “open region”).

Final Insight

Notice how the Maximum Modulus Principle guided the strategy:

• Instead of searching the interior,
• We immediately restricted attention to the boundary,
• Reducing the problem to real-variable calculus.

This illustrates the remarkable rigidity of analytic functions: their interior behavior is completely controlled by their boundary values.