Problem 3 - Complex Analysis

Question

Prove the "Triangle Inequality" $$ |z+w| \leq|z|+|w|, \quad z, w \in \mathbb{C} $$ and discuss when it becomes an equality; also prove the "Triangle Inequality" $$ || z|-| w|| \leq|z-w|, \quad z, w \in \mathbb{C} $$

Step-by-Step Solution

Answer

The inequality \(|z+w| \leq |z| + |w|\) holds with equality when \(z\) and \(w\) are collinear; \(|||z|-|w|| \leq |z-w|\) is shown by rearranging inequalities.

Step 1: Understand the Properties of Complex Numbers

Recall that for a complex number \(z = a + bi\), the modulus is given by \(|z| = \sqrt{a^2 + b^2}\). The modulus represents the distance from the origin to the point \((a, b)\) in the complex plane. Similarly, for \(w = c + di\), \(|w| = \sqrt{c^2 + d^2}\). The modulus operation satisfies \(|zw| = |z||w|\) and \(|z + w| eq |z| + |w|\) always, which sets the grounds for understanding inequalities involving complex numbers.

Step 2: Apply the Cauchy-Schwarz Inequality to Complex Numbers

Consider the vectors \(u = (a, b)\) and \(v = (c, d)\). The Cauchy-Schwarz inequality states that \(|u \cdot v| \leq ||u|| ||v||\). For the components \(u \cdot v = ac + bd\), and applying this in context of complex numbers, leads to proving \(|z+w|^2 \leq (|z|+|w|)^2\).

Step 3: Express \(|z+w|^2\) in Terms of \(z\) and \(w\)

For \(z = a + bi\) and \(w = c + di\), \(|z+w| = |(a+c) + (b+d)i| = \sqrt{(a+c)^2 + (b+d)^2}\). Calculate \(|z+w|^2 = (a+c)^2 + (b+d)^2\), which simplifies to \(a^2 + 2ac + c^2 + b^2 + 2bd + d^2\).

Step 4: Compare \(|z+w|^2\) and \((|z| + |w|)^2\)

Calculate \((|z| + |w|)^2 = (\sqrt{a^2 + b^2} + \sqrt{c^2 + d^2})^2 = a^2 + b^2 + c^2 + d^2 + 2(\sqrt{a^2 + b^2}\sqrt{c^2 + d^2})\). Recognize that since \(2ac + 2bd \leq 2(\sqrt{a^2 + b^2}\sqrt{c^2 + d^2})\) by Cauchy-Schwarz, \(|z+w|^2 \leq (|z|+|w|)^2\), proving \(|z+w| \leq |z| + |w|\).

Step 5: Determine Conditions for Equality

Equality holds in \(|z+w| \leq |z| + |w|\) when \(z\) and \(w\) are collinear vectors in the complex plane, meaning they have the same direction or one is a scalar multiple of the other.

Step 6: Prove Second Triangle Inequality \(|||z|-|w|| \leq |z-w|\)

By definition, \(|z-w| \geq ||z| - |w||\) can be shown by recognizing that \(|z| = |(z-w) + w|\). Applying previously proved \(|z+w| \leq |z| + |w|\) gives \(|z-w| + |w| \geq |z|\), rearranging gives \(|z-w| \geq ||z| - |w||\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are generally expressed in the form of \( z = a + bi \), where \( a \) is the real part, \( b \) is the imaginary part, and \( i \) is the imaginary unit that satisfies \( i^2 = -1 \).
The geometry of complex numbers can be visualized on the complex plane, where the x-axis denotes the real part and the y-axis denotes the imaginary part. This representation allows complex numbers to be treated like vectors.

• **Real Part**: The component without \( i \), denoted as \( a \).
• **Imaginary Part**: The component with \( i \), denoted as \( bi \).
• **Complex Conjugate**: If \( z = a + bi \), then its conjugate \( \bar{z} = a - bi \).

This dual nature of having both direction and magnitude through real and imaginary parts enables the rich interplay with moduli and vector-based operations which are key in understanding inequalities in complex number systems.

Modulus

The modulus of a complex number describes its distance from the origin on the complex plane. The modulus of \( z = a + bi \) is calculated as \( |z| = \sqrt{a^2 + b^2} \). This is similar to the magnitude of a vector in two-dimensional space.
The modulus has important properties that often simplify calculations with complex numbers:

• For any complex number \( z \), \(|z| \geq 0\), and \(|z| = 0\) if and only if \( z = 0 \).
• The modulus of the product of two complex numbers is the product of their moduli: \(|zw| = |z||w|\).
• The modulus of the sum satisfies the triangle inequality: \(|z + w| \leq |z| + |w|\).

Using the modulus helps in establishing inequalities, such as the Triangle Inequality, which is fundamental in many areas of mathematics, including complex analysis and geometry.

Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequality is a fundamental inequality that applies to vectors and sets the stage for various inequalities in mathematics. For vectors \( \mathbf{u} = (a, b) \) and \( \mathbf{v} = (c, d) \), the inequality is represented as:\[|u \cdot v| \leq ||u|| ||v|| \]
Where \( |u \cdot v| \) is the absolute value of the dot product of \( \mathbf{u} \) and \( \mathbf{v} \), and \( ||u|| \) and \( ||v|| \) are the magnitudes of \( \mathbf{u} \) and \( \mathbf{v} \), respectively.

• This inequality intuitively tells us that the absolute dot product of two vectors is maximized when they point in the same direction.
• It provides an upper bound for the magnitude of the dot product in terms of the magnitudes of both vectors.

This concept is particularly useful in complex analysis where it aids in establishing bounds for expressions like \(|z + w|^2\) in terms of \(|z|\) and \(|w|\), leading to proofs like the Triangle Inequality.

Collinear Vectors

Collinear vectors are vectors that lie along the same line, either in the same or opposite directions. In the complex plane, this translates to two complex numbers being scalar multiples of each other.
When vectors (or complex numbers) are collinear, the direction component aligns perfectly. In terms of the Triangle Inequality, equality \(|z + w| = |z| + |w|\) holds true when the vectors representing \(z\) and \(w\) are collinear.

• **Same Direction**: Vectors are parallel and move in the same direction; their scalar multiple is positive.
• **Opposite Direction**: Vectors are parallel but move in opposite directions; their scalar multiple is negative.
• This condition is essential when discussing when equality occurs in vector-related inequalities.

Understanding collinear vectors helps cement the conditions under which more complex inequalities become equalities, such as equality in the modulus-based Triangle Inequalities.