Complex Analysis: Continuity, Differentiability, and Limits

Functions and Objectives

(a) f(z) = |z|, where z is a complex number.

(b) f(x, y) = x²y / (x² + y²)

Proof Objective

• (a) Prove that f(z) = |z| is continuous everywhere but nowhere differentiable except at the origin.
• (b) Find the iterative limit and simultaneous limit of f(x, y) = x²y / (x² + y²) as (x, y) → (0, 0).

Proof Process

(a) Continuity of f(z) = |z|

[Step 1]: Show that f(z) = |z| is continuous everywhere.

Let z₀ be an arbitrary complex number. We want to show that for any ε > 0, there exists a δ > 0 such that if |z - z₀| < δ, then |f(z) - f(z₀)| < ε.

We have f(z) = |z| and f(z₀) = |z₀|. Then |f(z) - f(z₀)| = ||z| - |z₀||.

By the reverse triangle inequality, we know that ||z| - |z₀|| ≤ |z - z₀|.

So, if we choose δ = ε, then whenever |z - z₀| < δ, we have ||z| - |z₀|| ≤ |z - z₀| < δ = ε.

Differentiability of f(z) = |z|

[Step 2]: Show that f(z) = |z| is nowhere differentiable except at the origin.

Let z = x + iy, where x and y are real numbers. Then f(z) = |z| = √(x² + y²).

For f(z) to be differentiable at a point z₀, the limit lim_z→z₀ (f(z) - f(z₀)) / (z - z₀) must exist.

Let's consider the point z₀ = 0. Then we need to examine the limit lim_z→0 |z| / z. Let z = re^iθ. Then |z| = r, and z = re^iθ. So |z|/z = r / (re^iθ) = e^-iθ.

The limit lim_z→0 |z| / z = lim_r→0 e^-iθ. This limit depends on θ. For example, if θ = 0, the limit is 1. If θ = π/2, the limit is -i. Since the limit depends on the path taken to approach 0, the limit does not exist. Therefore, f(z) is not differentiable at z = 0.

Now, let's consider z₀ ≠ 0. We can write f(z) = √(z · conjugate(z)). If f(z) were differentiable at z₀, then ∂f/∂(conjugate(z)) = 0. However, ∂f/∂(conjugate(z)) = z / (2√(z · conjugate(z))) = z / (2|z|). This is not zero unless z = 0. Therefore, f(z) is not differentiable for z₀ ≠ 0.

(b) Iterative and Simultaneous Limits of f(x, y) = x²y / (x² + y²)

[Step 1]: Find the iterative limits.

First, we find the limit as x approaches 0, keeping y constant:

lim_x→0 x²y / (x² + y²) = 0 / y² = 0, for y ≠ 0.

Then, we find the limit as y approaches 0 of the result: lim_y→0 0 = 0.

So, the iterative limit lim_y→0 lim_x→0 x²y / (x² + y²) = 0.

Now, we find the limit as y approaches 0, keeping x constant:

lim_y→0 x²y / (x² + y²) = 0 / x² = 0, for x ≠ 0.

Then, we find the limit as x approaches 0 of the result: lim_x→0 0 = 0.

So, the iterative limit lim_x→0 lim_y→0 x²y / (x² + y²) = 0.

[Step 2]: Find the simultaneous limit.

We want to find lim_{(x, y)→(0, 0)} x²y / (x² + y²).

Let's use polar coordinates: x = r cos θ and y = r sin θ. Then x² + y² = r². f(x, y) = (r² cos² θ)(r sin θ) / r² = r cos² θ sin θ.

Now, we find the limit as r approaches 0: lim_r→0 r cos² θ sin θ = 0, since |cos² θ sin θ| ≤ 1.

Since the limit is 0 regardless of the value of θ, the simultaneous limit exists and is equal to 0.

Conclusion

(b) The iterative limits are both 0, and the simultaneous limit is also 0.

Definitions

• Harmonic Function: A twice continuously differentiable function that satisfies Laplace's equation: ∇²f = 0.
• Indexed Family of Sets: A collection of sets {Aᵢ} where each set is associated with an index i from an index set I.
• Maximal Element in a PO Set: An element in a partially ordered set that is not less than any other element.
• Lattice: A PO set in which every two elements have a least upper bound (join) and a greatest lower bound (meet).
• Diagonal Matrix: A square matrix where all entries outside the main diagonal are zero.
• Symmetric Matrix: A square matrix that is equal to its transpose, A = Aᵀ.

Additional Definitions

• Partition of an Interval: A finite set of points {x₀, x₁, ..., xₙ} such that a = x₀ < x₁ < ... < xₙ = b.
• Oscillatory Sum: Refers to the difference between the supremum and infimum of a function's values over a given interval.
• Simultaneous Limit: The limit of f(x, y) as (x, y) approaches (a, b) from any direction.
• Cauchy-Riemann Equations: The equations ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x, where f(z) = u(x, y) + iv(x, y).