Complex limits

Do, or do not. There is no try, Yoda.

Limits

In simple terms, the limit of a sequence {zₙ} is the value “z” that the terms of the sequence get arbitrarily close to as n (the index of the terms) becomes larger and larger (approaches infinity).

Formal Definition: Let {zₙ} be a sequence of complex numbers, and let z be a complex number. We say that the sequence {zₙ} converges to z, denoted or written as {zₙ} → z as n → ∞, if for every positive real number ε (epsilon: represents an arbitrary small positive number and defines how close the term zₙ need to be to the limit z, that is, the radius of a small circle, disk, or neighborhood centered at z), there exists a natural number N (a point in the sequence beyond which all subsequent terms fall into line, meaning they are all within the circle centered at z of radius ε) such that for all natural numbers n greater than or equal to N, the distance between zₙ and z is less than ε: ∀ε > 0, ∃N ∈ ℕ : (such that) ∀n ≥ N (“for all n greater than or equal to N”), |z_n - z| < ε.

In other words, no matter how small you choose ε (i.e., no matter how close you want the terms to be close to the limit, how small the disk or circle around z with radius ε is), you can always find a point in the sequence (N) such that all terms after that point lie inside this circle or are within a distance of ε from the limit z.

Example. Consider the sequence zₙ = (1 + i)/n, $\lim_{n \to \infty} \frac{1+i}{n} = 0$

∀ε > 0, ∃N ∈ ℕ, N = $\frac{2\sqrt{2}}{ε}$ : ∀n ≥ N, |z_n - z| < ε.

|z_n - z| = |(1 + i)/n| = |$\frac{1+i}{n}| = \frac{|1+i|}{n} = \frac{\sqrt{2}}{n} ≤ \frac{\sqrt{2}}{N} = \frac{ε}{2} < ε$

Theorem. Convergence of Complex Sequences and Real/Imaginary Parts A sequence of complex numbers {zₙ} converges to a complex number z if and only if the sequences of their real parts {xₙ} and imaginary parts {yₙ} converge to the real and imaginary parts of z, respectively. {zₙ} → z, where zₙ = xₙ + iyₙ ∀n, z = x + iy, if and only if {xₙ} → x and {yₙ} → y

Proof:

(=>)

Suppose $\lim_{n \to \infty} z_n = z$, for any ε > 0, there exists an N such that for all n ≥ N, |zₙ - z| < ε.

|zₙ - z| = |(xₙ - x) + i(yₙ - y)| = $\sqrt{((xₙ - x)² + (yₙ - y)²)}< ε => (xₙ - x)² < ε² \text{~and~} (yₙ - y)² < ε²$. Taking the square root, we get |xₙ - x| < ε and |yₙ - y| < ε.

This shows that for any ε > 0, there exists an N such that for all n ≥ N, |xₙ - x| < ε and |yₙ - y| < ε. Therefore, $\lim_{n \to \infty} x_n = x \text{~and~} \lim_{n \to \infty} y_n = y$.

(<==)

Assume $\lim_{n \to \infty} x_n = x \text{~and~} \lim_{n \to \infty} y_n = y$. This means for any ε > 0, there exist N₁ and N₂ such that for all n ≥ N₁, |xₙ - x| < ε/√2, and for all n ≥ N₂, |yₙ - y| < ε/√2.

Let N = max(N₁, N₂). Then for all n ≥ N, both inequalities hold, so now consider |zₙ - z| = $\sqrt{(xₙ - x)² + (yₙ - y)²} < \sqrt{\frac{ε²}{2} + \frac{ε²}{2}} = \sqrt{ε²} = ε$.

A Cauchy sequence is a sequence in which the terms become arbitrarily close to each other as the sequence progresses.

Formally, a complex sequence {zₙ} is called a Cauchy sequence if for every positive real number ε (indicating how close the terms should be), there exists a natural number N (indicating the point in the sequence after which the terms are guaranteed to be as close as required) such that for all natural numbers n and m greater than or equal to N, the distance between zₙ and zₘ is less than ε: ∀ε > 0, ∃N ∈ ℕ : (such that) ∀n, m ≥ N, |zₙ - zₘ| < ε.

Definition. A metric space is a set M equipped with a distance or metric function d: M × M → ℝ that defines a notion or construct of distance between any two points in the set. Basically, it satisfies the following four axioms. ∀x, y, z ∈ M (for any three points in the set):

1. Non-negativity: d(x, y) ≥ 0. The distance between any two points is non-negative.
2. Identity of indiscernibles: d(x, y) = 0 if and only if x = y. The distance between two points is zero exactly when they are the same point.
3. Symmetry: d(x, y) = d(y, x). The distance from x to y equals the distance from y to x.
4. Triangle inequality: d(x, z) ≤ d(x, y) + d(y, z). The distance from x to z is at most the sum of the distances from x to y and y to z.

Common Examples:

• (ℝ, d) where d(x, y) = |x - y|, the real line with usual distance.
• (ℝⁿ, d) where d(x, y) = $\sqrt{Σ(xᵢ - yᵢ)²}$ (Euclidean space).
• (ℂ, d) where d(z₁, z₂) = |z₁ - z₂|. The complex plane ℂ with the usual modulus as the distance function is a metric space.

Definition: A metric space (or a subset of a metric space) is said to be complete if every Cauchy sequence in that space (or subset) converges to a limit that is also within that space (or subset).

Consider the set of rational numbers ℚ (with the usual absolute value as the metric). The sequence defined by the decimal approximations of √2 (1, 1.4, 1.41, 1.414, …) is a Cauchy sequence in ℚ. However, the limit of this sequence is √2, which is not a rational number (√2 ∉ ℚ). Therefore, the set of rational numbers ℚ is not complete.

On the other hand, the set of real numbers ℝ (with the usual absolute value as the metric) is complete. Any Cauchy sequence of real numbers converges to a limit that is also a real number. Similarly, the complex plane ℂ is complete.

In summary: Completeness is a property of a metric space (or a subset thereof) that guarantees that Cauchy sequences within the space have limits that are also within the space.

Th. The complex plane is complete.

Proof.

Let {zₙ} be a Cauchy sequence in ℂ. This means that for any ε > 0, there exists an N ∈ ℕ such that for all n, m ≥ N, |zₙ - zₘ| < ε.

Express the complex numbers in terms of their real and imaginary parts: Let zₙ = xₙ + iyₙ and zₘ = xₘ + iyₘ, where xₙ, yₙ, xₘ, and yₘ are real numbers.

Rewrite the Cauchy condition: |zₙ - zₘ| = |(xₙ - xₘ) + i(yₙ - yₘ)| = $\sqrt{(xₙ - xₘ)² + (yₙ - yₘ)²}$ < ε

Since the square root of a sum of squares is less than ε, each square must also be less than ε²: (xₙ - xₘ)² < ε² and (yₙ - yₘ)² < ε²

Taking the square root of both sides gives: |xₙ - xₘ| < ε and |yₙ - yₘ| < ε

Cauchy sequences of real numbers: This shows that {xₙ} and {yₙ} are Cauchy sequences of real numbers. Since the real numbers ℝ are complete, every Cauchy sequence of real numbers converges to a real number. Therefore, there exist real numbers x and y such that: $\lim_{n \to \infty} xₙ = x \text{~and~} \lim_{n \to \infty} yₙ = y$

Next, we construct the complex limit: Let z = x + iy and aim to demonstrate that {zₙ} converges to z: We want to show that for any ε > 0, there exists an N such that for all n ≥ N, |zₙ - z| < ε.

|zₙ - z| = |(xₙ - x) + i(yₙ - y)| = $\sqrt{(xₙ - x)² + (yₙ - y)²}$

Since $\lim_{n \to \infty} xₙ = x \text{~and~} \lim_{n \to \infty} yₙ = y$, for any ε > 0, there exist N₁ and N₂ such that for all n ≥ N₁, |xₙ - x| < ε/√2, and for all n ≥ N₂, |yₙ - y| < ε/√2.

Let N = max(N₁, N₂). Then for all n ≥ N:

|zₙ - z| = $\sqrt{(xₙ - x)² + (yₙ - y)²} < \sqrt{(ε/√2)² + (ε/√2)²} = \sqrt{ε²/2 + ε²/2}$ = ε

Conclusion: We have shown that for any ε > 0, there exists an N such that for all n ≥ N, |zₙ - z| < ε. This means that the sequence {zₙ} converges to the complex number z = x + iy.

Proposition. We assume a ∈ ℂ, and f is a complex function with $\lim_{z \to a} f(z) = L_1$ and $\lim_{z \to a} f(z) = L_2$ then L₁ = L₂, in other words, limits are unique.

Proof.

Let ε > 0, there exists δ₁ and δ₂ for which 0 < | z -a | < δ₁ implies | f(z) - L₁ | < ε/2, and 0 < | z -a | < δ₂ implies | f(z) - L₂ | < ε/2

Let δ = min{δ₁, δ₂} and suppose 0 < | x -a | < δ ≤ δ₁, δ₂, then:

| L₁ - L₂ | = | L₁ - f(z) + f(z) - L₂ | ≤ | L₁ - f(z) | + | f(z) - L₂ | < ε/2 + ε/2 = ε. Thus, | L₁ - L₂ | < ε for arbitrary ε, that implies | L₁ - L₂ | = 0 ➞ L₁ = L₂.

Proposition. Limit of identity function, $\lim_{z \to a} z = a$.

Let ε > 0, there exists δ (choose δ = ε) for which 0 < | z -a | < δ = ε, obviously implies | z - a | < ε

Proposition. Limit of constant function, $\lim_{z \to a} c = c$.

Let ε > 0, there exists δ (choose δ = 42, the answer to the meaning of life, the universe and everything else) for which 0 < | z -a | < δ, obviously implies | c - c | = 0 < ε

Proposition. We assume a ∈ ℂ, and f and g are complex function with $\lim_{z \to a} f(z) = L_f$ and $\lim_{z \to a} g(z) = L_g$ then $\lim_{z \to a} f(z)+g(z) = L_f+L_g$.

Proof.

Let ε > 0, there exists δ₁ and δ₂ for which 0 < | z -a | < δ₁ implies | f(z) - L₁ | < ε/2, and 0 < | z -a | < δ₂ implies | g(z) - L₂ | < ε/2

Let δ = min{δ₁, δ₂} and suppose 0 < | x -a | < δ ≤ δ₁, δ₂, then:

$|f(z)+g(z) - (L_f+L_g)| = |f(z) - L_f + g(z) - L_g| ≤ | f(z) - L_f | + | g(z) - L_g | < ε/2 + ε/2 = ε $

Proposition. Homogeneity of the limit. Suppose c ∈ ℂ and $\lim_{z \to a} f(z) = L$, then $\lim_{z \to a} cf(z) = c\lim_{z \to a} f(z)$

Assume c ≠ 0. Let ε > 0, there exists δ for which 0 < | z -a | < δ implies | f(z) - L | < ε/|c|

| cf(z) - cL | = |c|∙| f(z) - L | < |c| ∙ ε/|c| = ε.

If c = 0, | 0f(z) - 0L | = 0 < ε for all z ∈ dom(f).

Proposition. Limit of linear combination of convert functions. Suppose a ∈ ℂ, and $\lim_{z \to a} f_i(z) = L_i$. Then, $\lim_{z \to a} (cf_1+cf_2 + … + cf_n)(z) = c_1\lim_{z \to a} f_1(z) + c_2\lim_{z \to a} f_2(z) + … + c_n\lim_{z \to a} f_n(z) $

Proof. We will prove it by induction on n.

n = 1. It was previously proved. Homogeneity of the limit, $\lim_{z \to a} cf(z) = c\lim_{z \to a} f(z)$.

Inductively, suppose the claim is true for some n ∈ ℕ, consider the limit of the linear combination of $\lim_{z \to a} (cf_1+cf_2 + … + cf_{n+1})(z) = \lim_{z \to a} (cf_1+cf_2 + … + cf_{n})(z)+ \lim_{z \to a} cf_n(z) = c_1\lim_{z \to a} f_1(z) + c_2\lim_{z \to a} f_2(z) + … + c_n\lim_{z \to a} f_n(z) + c_{n+1}\lim_{z \to a} f_{n+1}(z)$

Proposition. Limit of products is product of limits. We assume a ∈ ℂ, and f and g are complex function with $\lim_{z \to a} f(z) = L_f$ and $\lim_{z \to a} g(z) = L_g$ then $\lim_{z \to a} f(z)·g(z) = L_f·L_g$.

Proof

$|f(z)·g(z) - L_f·L_g| = |f(z)g(z) -L_fg(z) + L_fg(z) - L_fL_g| ≤ |f(z) -L_f||g(z)| + |L_f||g(z) - L_g|$

Let ε > 0, there exists δ₁ for which 0 < | z -a | < δ₁ implies $| f(z) - L_f | < \frac{ε}{2(1+|L_g|)}$. Analogously, there exists δ₂ for which 0 < | z -a | < δ₂ implies $| g(z) - L_g | < \frac{ε}{2(1+|L_f|)}$. Besides, there exists δ₃ for which 0 < | z -a | < δ₃ implies $| g(z) - L_g | < 1 ⇒ | g(z) | = | g(z) - L_g + L_g | ≤ | g(z) - L_g | + | L_g | < 1 + | L_g |$

Let δ = min{ δ₁, δ₂, δ₃}, 0 < | z -a | < δ:

$|f(z)·g(z) - L_f·L_g| ≤ |f(z) -L_f||g(z)| + |L_f||g(z) - L_g| < \frac{ε}{2(1+|L_g|)}(1 + | L_g |) + |L_f|\frac{ε}{2(1+|L_f|)} < \frac{ε}{2}+\frac{ε}{2} = ε$ because $\frac{|L_f|}{1+|L_f|} < 1$

Proposition. Power function limit. Let a ∈ ℝ and n ∈ ℕ, $\lim_{x \to a} z^n = a^n$

Proof by induction on n.

n = 0, $\lim_{x \to a} 1 = 1$. It has already been proved by Proposition. Limit of constant function, $\lim_{z \to a} c = c$ where c = 1.

Inductively, suppose that $\lim_{x \to a} z^n = a^n$, then:

$\lim_{x \to a} z^{n+1}=\lim_{x \to a} z^n·z =[\text{By product of limit theorem}] (\lim_{x \to a} z^n)·(\lim_{x \to a} z) =[\text{By induction and limit of identity function}] a^n·a = a^{n+1}.$

Proposition. Polynomial function limit. Suppose $c_0, c_1, c_2, ···, c_n$ ∈ ℝ, and p(z) = $c_0 + c_1z + c_2z² + ··· + c_nz^n$, then $\lim_{x \to a}p(z) = p(a)$ (direct consequence of previous results).

Proposition. Reciprocal function limit. Let a ∈ ℝ, a ≠ 0, and n ∈ ℕ, $\lim_{z \to a} \frac{1}{z} = \frac{1}{a}$

Proof.

$\frac{1}{z} = \frac{x-iy}{x²+y²} = \frac{x}{x²+y²} + \frac{-iy}{x²+y²} = u(z) + iv(z)$.

Let $a = x_0 + iy_0$, a ≠ 0, then since the functions are continuous at a (Calculus multivariable), $x_0²+y_0² > 0, \lim_{z \to a}u(z) = \frac{x_0}{x_0²+y_0²}, \lim_{z \to a}u(z) = \frac{x_0}{x_0²+y_0²}, \lim_{z \to a}v(z) = \frac{-y_0}{x_0²+y_0²}$

$\lim_{z \to a} \frac{1}{z} = \frac{x_0}{x_0²+y_0²} + i\frac{-y_0}{x_0²+y_0²} = \frac{x_0-iy_0}{x_0²+y_0²} = \frac{1}{a}$

Proposition. Limit of quotient is quotient of limits. We assume a ∈ ℂ, and f and g are complex function with $\lim_{z \to a} f(z) = L_f$ and $\lim_{z \to a} g(z) = L_g$ with L_g ≠ 0 then $\lim_{z \to a} \frac{f(z)}{g(z)} = \frac{L_f}{L_g}$.

Differentiation on Euclidean Space

If F : U ⊆ ℝⁿ ➞ ℝⁿ is a function, and a ∈ U is an interior point of U for which the limit $\lim_{h \to 0} \frac{F(a+h)-F(a)-L(h)}{||h||}$ exists where L(h) = DF_a is a linear transformation, then we say F is differentiable at a with derivative Df_a: ℝⁿ ➞ ℝⁿ

1. Domain and Codomain: F maps from a subset U of ℝⁿ (n-dimensional Euclidean space) to ℝᵐ (m-dimensional Euclidean space). So, the input is a vector in ℝⁿ and the output is a vector in ℝᵐ.
2. Interior Point: a must be an interior point of U. This means there exists an open n-dimensional disk, ball or neighborhood centered at “a” that is entirely contained within U. This is necessary because we need to consider F(a+h) for small values of h, and we need a+h to be within the domain of f.
3. The Limit: The crucial part is the limit: $\lim_{h \to 0} \frac{F(a+h)-F(a)-L(h)}{||h||}$ where h is a vector in ℝⁿ representing a small displacement from a. The limit is taken as h approaches the zero vector in ℝⁿ. F(a+h) - F(a) is the difference between the function’s value at a+h and its value at a. This is a vector in ℝᵐ.
4. L(h) = Dfₐ(h): This is a linear transformation from ℝⁿ to ℝᵐ. It represents the best linear approximation of the change in F near a. Dfₐ is called the derivative (or differential) of f at a. ||F(a+h) - F(a) - L(h)|| is the norm (or magnitude) of the vector F(a+h) - F(a) - L(h). It measures the “error” of the linear approximation. This is a real number.
5. ||h||: This is the norm (or magnitude) of the vector h. This is also a real number.
6. The fraction $\frac{F(a+h)-F(a)-L(h)}{||h||}$ represents the relative error of the linear approximation. Its limit equal to 0, meaning that the relative error goes to zero as h approaches zero. This is what makes L the “best” linear approximation.

The Derivative DFₐ: The derivative DFₐ is a linear transformation, not just a value. It’s often represented by a matrix (the Jacobian matrix) when working with functions from ℝⁿ to ℝᵐ.

Let e_j = (0, 0, ···1_j···, 0), (e_j)_i defined as

$(e_j)_i = \begin{cases} 0, i ≠ j \\ 1, i = j \end{cases}$

The identity matrix I = [e₁, e₂, ···, e_n]

Definition. Let F : U ⊆ ℝⁿ → ℝᵐ be a function, where U is an open set in ℝⁿ. Let a = (a₁, a₂, …, aₙ) be a point in U. Let eⱼ be the j-th standard basis vector in ℝⁿ. The partial derivative of F with respect to xⱼ at the point a, denoted by $\frac{∂F}{∂x_j}$ is defined as $\frac{∂F}{∂x_j}(a) = \lim_{t \to 0} [\frac{F(a+te_j)-F(a)}{t}]$ where

• F(a + teⱼ) - F(a) is the change in the function’s value as we move a small distance t in the j-th direction.
• $\frac{F(a+te_j)-F(a)}{t}$ represents the average rate of change of F with respect to xⱼ over the interval [aⱼ, aⱼ + t].
• The limit as t → 0 gives us the instantaneous rate of change of F with respect to xⱼ at the point a.

Example. F(s, t) = (s²+t³, 2st, s + 3t), F : ℝ² ➞ ℝ³

$\frac{∂F}{∂s} = (\begin{smallmatrix}2s\\ 2t\\ 1\end{smallmatrix}), \frac{∂F}{∂t} = (\begin{smallmatrix}3t²\\ 2s\\ 3\end{smallmatrix})$.

The Jacobian matrix J_F (or the differential dF_(s,t)) is formed by placing these column vectors side by side: $J_F =[\frac{∂F}{∂s}|\frac{∂F}{∂t}] = [dF_{s, t}] = (\begin{smallmatrix}2s & 3t²\\ 2t & 2s\\ 1 & 3\end{smallmatrix})$

The definition of the partial derivative is the limit of the difference quotient as the change in the corresponding variable approaches zero.

Definition. Let F : U ⊆ ℝⁿ → ℝᵐ be a function, where U is an open set in ℝⁿ. Suppose F can be written in terms of its component functions: F(x) = (F₁(x), F₂(x), …, Fₘ(x)) where each Fᵢ : U ⊆ ℝⁿ → ℝ is a scalar-valued function.

The Jacobian matrix of F at a point a ∈ U, denoted by J_F(a), is an m × n matrix whose entries are the partial derivatives of the component functions: $J_F =[\text{the Jacobian matrix can be written in terms of column vectors (partial derivatives with respect to each variable)}] [\frac{∂F}{∂x_1}|\frac{∂F}{∂x_2}| ··· |\frac{∂F}{∂x_n}] = (\begin{smallmatrix}\frac{∂F_1}{∂x_1} & \frac{∂F_1}{∂x_2} & ··· & \frac{∂F_1}{∂x_n}\\\frac{∂F_2}{∂x_1} & \frac{∂F_2}{∂x_2} & ··· & \frac{∂F_2}{∂x_n}\\· & · & · & ·\\· & · & · & ·\\· & · & · & ·\\\frac{∂F_m}{∂x_1} & \frac{∂F_m}{∂x_2} & ··· & \frac{∂F_m}{∂x_n}\end{smallmatrix}) = (\begin{smallmatrix}(∇F_1)^T\\ ··· \\ (∇F_2)^T\\ (∇F_m)^T\end{smallmatrix})$ where $(∇F_i)^T$ represents the transpose of the gradient vector of Fᵢ at a

Theorem. If F : U ⊆ ℝⁿ → ℝᵐ is differentiable at a point a ∈ U, then the differential dFₐ(h) (which is a linear transformation from ℝⁿ to ℝᵐ) can be represented by matrix multiplication using the Jacobian matrix:

dFₐ(h) = J_F(a)·h where h is a vector in ℝⁿ (represented as a column vector).

The differential dFₐ is the best linear approximation of the change in F near the point a. The Jacobian matrix J_F(a) is the matrix that represents this linear transformation. When you multiply the Jacobian matrix by the vector h, you get the approximate change in F corresponding to the small change h.

Examples:

• F(s, t) = (s²+t³, 2st, s + 3t), F : ℝ² ➞ ℝ³, $\frac{∂F}{∂s} = (\begin{smallmatrix}2s\\ 2t\\ 1\end{smallmatrix}), \frac{∂F}{∂t} = (\begin{smallmatrix}3t²\\ 2s\\ 3\end{smallmatrix}), J_F = (\begin{smallmatrix}2s & 3t²\\ 2t & 2s\\ 1 & 3\end{smallmatrix}), J_F(1, 2) = (\begin{smallmatrix}2 & 12\\ 4 & 2\\ 1 & 3\end{smallmatrix})$.
• f(z) = e^z = e^xcos(y) + ie^xsin(y). $\frac{∂f}{∂x} = \frac{∂}{∂x}(e^xcos(y))+ i\frac{∂}{∂x}(e^xsin(y)) = e^xcos(y) + ie^xsin(y) = e^z, \frac{∂f}{∂y} = \frac{∂}{∂y}(e^xcos(y))+ i\frac{∂}{∂y}(e^xsin(y)) = -e^xsin(y) + ie^xcos(y) = ie^z, J_f = [e^z | ie^z]$
• g(z) = z² = (x +iy)(x -iy) = x² -y² +2ixy. $\frac{∂g}{∂x} = \frac{∂}{∂x}(x²-y²)+ i\frac{∂}{∂x}(2xy) = 2x + i2y = 2z, \frac{∂g}{∂y} = \frac{∂}{∂y}(x²-y²)+ i\frac{∂}{∂y}(2xy) = -2y + i(2x) = 2iz, J_g = [z | 2iz]$

Theorem. Suppose f, g: ℂ ➞ ℂ have partial derivatives which exist then:

1. $∂_x(f±g) = ∂_x(f) ± ∂_x(g), ∂_y(f±g) = ∂_y(f) ± ∂_y(g)$
2. $∂_x(fg) = (∂_x(f))g + f(∂_x(g)), ∂_y(fg) = (∂_y(f))g + f(∂_y(g))$ (Product rule)

Proof (Product rule)

Suppose f = u + iv, g = a + ib, fg = (u + iv)(a + ib) = ua -ub + i(ub +va)

Now, let’s take the partial derivatives:

∂/∂x(fg) = ∂/∂x(ua - vb) + i∂/∂x(ub + va) …

… = (∂u/∂x)a + u(∂a/∂x) - (∂v/∂x)b - v(∂b/∂x) + i[(∂u/∂x)b + u(∂b/∂x) + (∂v/∂x)a + v(∂a/∂x)] þ*|

Now, if we assume that a and b are the real and imaginary parts of a function g(z) = a(x,y) + ib(x,y), then we can use the product rule for real valued functions.

If a and b are independent of x (i.e., they are constants or functions of y only), then ∂a/∂x = 0 and ∂b/∂x = 0. In this simpler case, the expression becomes:

∂/∂x(fg) = þ*| (∂u/∂x)a - (∂v/∂x)b + i[(∂u/∂x)b + (∂v/∂x)a]

= (∂u/∂x + i∂v/∂x) (a + ib) = (∂/∂x)f g

If we consider the case where a and b are not independent of x, we can rewrite the expression as:

∂/∂x(fg) = þ*| (∂u/∂x + i∂v/∂x)(a+ib) + (u+iv)(∂a/∂x+i∂b/∂x) =[f = u + iv, g = a + ib] (∂f/∂_x)g + f(∂g/∂_x)

The proof of $∂_y(fg) = (∂_y(f))g + f(∂_y(g))$ is completely similar.

Exercise

• sin(z) = sin(x + iy) =[Angle sum formula for sine, sin(A + B) = sin(A)cos(B) + cos(A)sin(B)] sin(x)cos(iy) + cos(x)sin(iy) =[Since cos(iy) = cosh(y) and sin(iy) = i·sinh(y)] sin(x)cosh(y) + icos(x)sinh(y)
• cos(z) = cos(x +iy) =[Angle sum formula for cosine, cos(A + B) = cos(A)cos(B) - sin(A)sin(B)] cos(x)cos(iy) - sin(x)sin(iy) =[Using the same substitutions as above: cos(iy) = cosh(y) and sin(iy) = i·sinh(y)] cos(x)cosh(y) -isin(x)sinh(y)
• ∂_xsin(z) =[Derivatives of real trigonometric functions: ∂_xsin(x) = cos(x), ∂_xcos(x) = -sin(x) and consider the fact that cosh(y) and sinh(y) are independent of x, so they are treated as constants when differentiating with respect to x] cos(x)cosh(y) -isin(x)sinh(y) = cos(z)
• ∂_xcos(z) = -sin(x)cosh(y) +icos(x)sinh(y) = -sin(z).
• z² = (x + iy)² = x² -y² +2xyi. Applying the formula for sin(z) and cos(z): sin(z²) = sin(x² -y²)cosh(2xy) + icos(x² -y²)sinh(2xy), cos(z²) = cos(x² -y²)cosh(2xy) -isin(x² -y²)sinh(2xy)
• ∂_xsin(z²) =[Chain Rule: d/dx f(g(x)) = f’(g(x)) * g’(x), d/dx sinh(x) = cosh(x), d/dx cosh(x) = sinh(x)] 2xcos(x² -y²)cosh(2xy) +2ysin(x² -y²)sinh(2xy) + i[-2xsin(x² -y²)sinh(2xy) +2ycos(x² -y²)cosh(2xy)] = 2x[cos(x² -y²)cosh(2xy) -isin(x² -y²)sinh(2xy)] + 2y[icos(x² -y²)cosh(2xy) + sin(x² -y²)sinh(2xy)] = 2xcos(z²)+2iycos(z²) = 2(x + iy)cos(z²) = 2zcos(z²).

Examples of Jacobian Matrices

• Suppose f(z) = z² defines a function or mapping on ℂ. If z = x + iy, then z² = (x + iy)² = (x² -y², 2xy). $J_f = [∂_xf | ∂_yf] =(\begin{smallmatrix}2x & -2y\\ 2y & 2x\end{smallmatrix})$
• Let f(x, y) = (x² + y², 2xy), $J_f = [∂_xf | ∂_yf] =(\begin{smallmatrix}2x & 2y\\ 2y & 2x\end{smallmatrix})$
• Let f(x, y) = (x² + y², xy, x + y), the Jacobian is a 3 x 2 matrix: $J_f = [∂_xf | ∂_yf] =(\begin{smallmatrix}2x & 2y\\ y & x\\ 1 & 1\end{smallmatrix})$
• Let $f(r, \theta) = (rcos(\theta), rsin(\theta))$, the Jacobian is a 2 x 2 square matrix, $J_f = [∂_rf | ∂_θf] = (\begin{smallmatrix} cos(\theta) & -rsin(\theta)\\ sin(\theta) & rcos(\theta)\end{smallmatrix})$
• Let f(z) = e^z where z = x + iy ∈ ℂ, we know that e^z = e^x(cosy + isin(y)), therefore $J_f = [∂_xf | ∂_yf] =(\begin{smallmatrix}e^xcos(y) & -e^xsin(y)\\ e^xsin(y) & e^xcos(y)\end{smallmatrix})$
• Let f(z) =$3z + \bar z$ where z = x + iy, $\bar z = x -iy, f(x, y) = (4x, 2y), J_f = [∂_xf | ∂_yf] =(\begin{smallmatrix}4 & 0\\ 0 & 2\end{smallmatrix})$
• Let f be the dot product of two vectors in ℝ³ (notice that the result of the dot product is a scalar, a real number) $f(\vec{x},\vec{y})=\vec{x}·\vec{y} = x_1y_1, x_2y_2, x_3y_3$, f is a mapping from f: ℝ³ x ℝ³ ➞ ℝ.

The differential of a function at a point $(\vec{x},\vec{y})$, denoted by $df_{(\vec{x},\vec{y})}$ is a linear transformation that best approximates the change in f for small changes in its arguments. Because our input is ($\vec{x}$, $\vec{y}$) ∈ ℝ³xℝ³, a small change around that point will be ($Δ\vec{x}$, $Δ\vec{y}$). The differential $df_{(\vec{x},\vec{y})}$ is a linear transformation in these increments $Δ\vec{x}$ and $Δ\vec{y}$.

To find the matrix (or row-vector) representation of the differential, we compute the gradient of f, treating $\vec{x} = (x_1, x_2, x_3)$ and $\vec{y} = (y_1, y_2, y_3)$ components as independents variables. Hence, f depends on six independent variables. To find the differential, we can compute the partial derivatives of f with respect to each component of $\vec{x}$ and $\vec{y}$

$\frac{∂f}{∂x_1} = y_1, \frac{∂f}{∂x_2} = y_2, \frac{∂f}{∂x_3} = y_3, \frac{∂f}{∂y_1} = x_1, \frac{∂f}{∂y_2} = x_2, \frac{∂f}{∂y_3} = x_3, ∇f(\vec{x},\vec{y}) = \Biggl(\begin{smallmatrix}y_1\\ y_2\\ y_3 \\x_1 \\x_2 \\x_3\end{smallmatrix}\Biggr)$

$[df_{(\vec{x},\vec{y})}] = ∇f(\vec{x},\vec{y})^T = [y_1, y_2,y_3,x_1,x_2,x_3]$

Notice that if we have small changes or increments in $\vec{x}$ and $\vec{y}$, denoted by $Δ\vec{x}$ = (Δx₁, Δx₂, Δx₃) and $Δ\vec{y}$ = (Δy₁, Δy₂, Δy₃), then the approximate change in f is given by: $df_{(\vec{x},\vec{y})}(Δ\vec{x}, Δ\vec{y})$ and found by taking the dot product of $∇f(\vec{x},\vec{y})$ with the increment vector: $df_{(\vec{x},\vec{y})}(Δ\vec{x}, Δ\vec{y}) = [y_1, y_2,y_3,x_1,x_2,x_3] * (Δx_1, Δx_2, Δx_3, Δy_1, Δy_2, Δy_3)^T = y_1Δx_1 + y_2Δx_2 + y_3Δx_3 + x_1Δy_1 + x_2Δy_2 + x_3 = (\vec{y}·Δ\vec{x}) + (\vec{x}·Δ\vec{y})$ where $(\vec{y}·Δ\vec{x})$ is the change contributed by $Δ\vec{x}$ while $\vec{y}$ is held fixed, and analogously $(\vec{x}·Δ\vec{y})$ is the change contributed by $Δ\vec{y}$ while $\vec{x}$ is held fixed. Thus, the total small change is $f(\vec{x}, \vec{y}) = \vec{x} · \vec{y}$ is the sum of these two small changes, and this argument perfectly aligns with our intuition about how the dot product operates.