Question

Show that a complex function \(f: \mathbb{C} \supset \Omega \rightarrow \mathbb{C}\) considered as a map \(f: \mathbb{R}^{2} \supset \Omega \rightarrow \mathbb{R}^{2}\) is differentiable iff it satisfies the Cauchy-Riemann conditions. Hint: Consider the Jacobian matrix of \(f\), and note that a linear complex map \(\mathbf{T}: \mathbb{C} \rightarrow \mathbb{C}\) is necessarily of the form \(\mathbf{T}(z)=\lambda z\) for some constant \(\lambda \in \mathbb{C} .\)

Short answer

The function \(f\) is differentiable if and only if it satisfies the Cauchy-Riemann equations. The proof involves showing that the Jacobian matrix of a differentiable function must satisfy the Cauchy-Riemann conditions, and that any function that satisfies the Cauchy-Riemann conditions can be expressed as a linear map and is therefore differentiable.

Step-by-step solution

Understand the Problem

We are given a complex function and the task is to prove that it is differentiable if and only if it satisfies the Cauchy-Riemann conditions. We're guided to consider the Jacobian matrix of the function, which is a key concept in multivariable calculus and differential equations. The hint infers that this problem may be approached by considering the properties of a linear complex map.

Proving the Forward Direction

The forward direction to prove is: If a function is differentiable, then it satisfies the Cauchy-Riemann conditions. Assume \(f\) is differentiable at \(z\). Then the derivative \(f'\) exists and by definition of the complex derivative \(f'(z)\), it can be expressed in terms of the Jacobian matrix as \(df/dz = [u'_x - v'_y, u'_y + v'_x]\). Since \(f\) is differentiable, the Jacobian matrix should exist and be equal to \(f'(z)\), i.e. \([u'_x - v'_y = f'(z), u'_y + v'_x = 0]\). From these, we obtain the Cauchy-Riemann conditions, \(u'_x = v'_y\) and \(u'_y = -v'_x\).

Proving the Reverse Direction

Next, we need to prove the reverse direction: If a function satisfies the Cauchy-Riemann equations, then it is differentiable. Assume \(f\) satisfies the Cauchy-Riemann equations. Then, the Jacobian matrix of \(f\) is of the form: \([u'_x, -v'_x; v'_x, u'_x]\) which satisfies the condition for the Jacobian of a differentiable function. That means \(f\) can be expressed as a linear map \(f(z) = \lambda z\), and so \(f\) is differentiable.

Key concepts

Cauchy-Riemann Conditions

One of the foundational elements in complex analysis is the set of equations known as the Cauchy-Riemann conditions. These are crucial for determining the differentiability of a function in the context of complex numbers.

For a complex function, expressed in terms of real variables as , where represents the real part and indicates the imaginary part, the Cauchy-Riemann conditions are represented by the following equations:----These conditions assert that partial derivatives of the real and imaginary components of must satisfy certain relationships. When a complex function fulfills these conditions at a certain point, it's a strong indicator that the function is complex differentiable (or holomorphic) at that point.

It's important to recognize that satisfying the Cauchy-Riemann conditions is necessary for differentiability but not sufficient on its own. For sufficiency, the partial derivatives also need to be continuous. This nuanced requirement is essential for students to grasp, avoiding any misunderstanding that these conditions alone guarantee differentiability.

Jacobian Matrix

The Jacobian matrix is a powerful tool in multivariable calculus, especially when it comes to understanding differentiable mappings in higher dimensions. When a complex function of a complex variable is considered, the Jacobian matrix takes on a very specific and useful form.

In complex analysis, the Jacobian matrix of a function can be seen as a bridge between complex differentiation and multivariable real differentiation. For a complex function presented as , the Jacobian matrix is a 2x2 matrix that contains all the first-order partial derivatives. It looks like this:----This matrix represents the local linear approximation of at point . When a complex function is differentiable at a point, its Jacobian matrix will conform to certain patterns that mirror the Cauchy-Riemann conditions. Specifically, for a complex differentiable function, the Jacobian must be of the form:----This structure is what allows for the correlation between the Jacobian matrix and the Cauchy-Riemann conditions. Understanding the Jacobian matrix provides students with a clearer conceptual understanding of differentiability in complex analysis.

Differentiable Maps in Complex Analysis

In complex analysis, a map is considered differentiable, or holomorphic, if it satisfies certain criteria. A differentiable map isn't just smoothly varying; it carries complex structure from one complex plane to another.

For a map to be differentiable at a point , it needs to have a derivative at that point in the complex sense. This means that the function can be approximated by a linear function at that point, and the behavior of the function near can be predicted by that linear approximation. The Cauchy-Riemann conditions emerge as part of the criteria for such differentiability because they ensure the existence of a complex derivative.

When exploring differentiable maps in complex analysis, it's also important to remember that complex differentiability has stricter requirements than real differentiability. A function could very well have real derivatives but fail to be complex differentiable if it doesn't adhere to the conditions required for complex structures. This distinction underscores the unique nature of complex analysis compared to its real counterpart.

Linear Complex Maps

Linear complex maps are a type of transformation in complex analysis where every point in the complex plane is translated uniformly according to a certain rule. A linear transformation in the complex plane can be expressed in the general form , where is a constant complex number.

What makes linear maps particularly interesting is their simplicity and their role in evaluating the differentiability of more complex functions. If a function behaves locally like a linear map, that characteristic supports the function being complex differentiable at that point. Moreover, the hint given in the exercise suggests the direct correlation between linear complex maps and differentiability, indicating the linear nature of the derivative.

Understanding linear complex maps helps students to visualize and comprehend the concept of complex differentiability more clearly, by elucidating how complex functions should behave locally around differentiable points. It reinforces the connection between the algebraic structure of complex numbers and the geometric interpretation of functions on the complex plane.