Question

Suppose \(f\) and \(g\) are complex differentiable functions on \((0,1), f(x) \rightarrow 0, g(x) \rightarrow 0\), \(f^{\prime}(x) \rightarrow A, g^{\prime}(x) \rightarrow B\) as \(x \rightarrow 0\), where \(A\) and \(B\) are complex numbers, \(B \neq 0 .\) Prove that $$ \lim _{x \rightarrow 0} \frac{f(x)}{g(x)}=\frac{A}{B} $$ Compare with Example 5.18. Hint: $$ \frac{f(x)}{g(x)}=\left\\{\frac{f(x)}{x}-A\right\\} \cdot \frac{x}{g(x)}+A \cdot \frac{x}{g(x)} $$ Apply Theorem \(5.13\) to the real and imaginary parts of \(f(x) / x\) and \(g(x) / x\).

Short answer

Question: Prove that if complex differentiable functions \(f\) and \(g\) on \((0,1)\) have limits \(\lim_{x\to 0} \frac{f(x)}{x} = A\) and \(\lim_{x\to 0} \frac{g(x)}{x} = B\) (with \(B \neq 0\)), then \(\lim_{x\to 0} \frac{f(x)}{g(x)} = \frac{A}{B}\). Answer: We proved that \(\lim_{x\to 0} \frac{f(x)}{g(x)} = \frac{A}{B}\) by rewriting the expression using the given hint, taking the limit as \(x\) approaches \(0\), applying Theorem 5.13 to the real and imaginary parts of \(\frac{f(x)}{x}\) and \(\frac{g(x)}{x}\), calculating the limit of the remaining terms, and combining the limit results.

Step-by-step solution

Rewrite the expression using the hint

Given the hint, rewrite \(\frac{f(x)}{g(x)}\) as: $$ \frac{f(x)}{g(x)} = \left\{\frac{f(x)}{x} - A\right\} \cdot \frac{x}{g(x)} + A \cdot \frac{x}{g(x)} $$

Take the limit as \(x\) approaches \(0\)

We want to find \(\lim_{x \rightarrow 0} \frac{f(x)}{g(x)}\). With the rewritten expression, this limit can be given as: $$ \lim_{x \rightarrow 0} \frac{f(x)}{g(x)} = \lim_{x \rightarrow 0} \left\{\left(\frac{f(x)}{x} - A\right) \cdot \frac{x}{g(x)} + A \cdot \frac{x}{g(x)}\right\} $$

Apply Theorem 5.13 to the real and imaginary parts of \(\frac{f(x)}{x}\) and \(\frac{g(x)}{x}\)

Theorem 5.13 states that if a function is differentiable at a point, then the real and imaginary parts are both differentiable at that point, and the derivative can be found by differentiating each part separately. Let the real part of \(\frac{f(x)}{x}\), \(R_1(x)\), and imaginary part, \(I_1(x)\), as \(x \rightarrow 0\) be \(R_1 \rightarrow A_R\) and \(I_1 \rightarrow A_I\) respectively. Similarly, let the real and imaginary parts of \(\frac{g(x)}{x}\), \(R_2(x)\) and \(I_2(x)\), as \(x \rightarrow 0\) be \(R_2 \rightarrow B_R\) and \(I_2 \rightarrow B_I\) respectively. Since \(f\) and \(g\) are differentiable functions on \((0,1)\), we can apply Theorem 5.13 to find: $$ \lim_{x \rightarrow 0} \left\{\left(\frac{f(x)}{x} - A\right) \cdot \frac{x}{g(x)}\right\} = \lim_{x \rightarrow 0} (R_1(x) - A_R + i(I_1(x) - A_I))(R_2(x) + i I_2(x)) = 0 $$

Calculate the limit of the remaining terms

Now, we need to calculate the limit of the other term in the provided expression: $$ \lim_{x \rightarrow 0} A \cdot \frac{x}{g(x)} = A \cdot \lim_{x \rightarrow 0} \frac{x}{g(x)} = A \cdot \lim_{x \rightarrow 0} (R_2(x) + i I_2(x)) = A \cdot \frac{1}{B} $$

Combine the limit results

Using our results from Step 3 and 4, we can now find the final limit of \(\frac{f(x)}{g(x)}\): $$ \lim_{x \rightarrow 0} \frac{f(x)}{g(x)} = 0 + A \cdot \frac{1}{B} = \frac{A}{B} $$ Thus, we have proved the required expression: $$ \lim_{x \rightarrow 0} \frac{f(x)}{g(x)} = \frac{A}{B} $$

Key concepts

Complex Differentiability

In complex analysis, differentiability is a cornerstone concept that extends beyond the real numbers to the more complex plane. When a function of a complex variable, such as \(f(z)\), is differentiable at a point, it implies that not only can you compute a derivative at that point, but also that the behavior of the function is locally linear. This means around that point, the function behaves almost like a straight-line function. It's important to note that for a complex function to be differentiable at a point, it must also satisfy the Cauchy-Riemann equations, connecting the real and imaginary parts.

• **Real and Imaginary Parts**: For the function \(f(z) = u(x, y) + iv(x, y)\), both \(u(x, y)\) and \(v(x, y)\) must have partial derivatives that match specific conditions.
• **Local Behavior**: Successful differentiation in the complex plane ensures that the function behaves predictably around a specific point.
• **Cauchy-Riemann Equations**: These equations are \(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\) and \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\).

If a function lacks differentiability at any point in a region, we cannot apply the same tools of calculus effectively.

Limits in Complex Analysis

Limits serve as the foundation of many concepts in calculus, providing a way to understand the behavior of functions as inputs approach a specific point. In complex analysis, limits help us understand how functions behave as the complex variable \(z\) approaches a specific value. For any limit like \(\lim_{z \to a} f(z) = L\), the function \(f(z)\) gets arbitrarily close to \(L\) as \(z\) gets closer to \(a\).

• **Path Independence**: In the complex plane, the limit must be consistent regardless of the path taken to approach \(z = a\). This is a key difference from real analysis.
• **Sequential Criterion**: If a limit exists, it must also hold for sequences approaching the point within the complex plane.

Understanding limits in complex analysis is crucial for discussing continuity and differentiability. A function needs to have a limit to be continuous, and continuity is a requirement for differentiability.

Division of Complex Functions

Dividing complex functions is a task that requires careful handling, especially in scenarios involving limits and complex differentiability. When dealing with functions \(f(z)\) and \(g(z)\) that both approach 0, as in the given exercise, a direct division might lead to an indeterminate form. To resolve this:

• **Apply Limits**: Use limits to simplify the expression. In this case, transforming the division into a form that isolates the derivatives of the functions allows you to utilize the stabilization at \((0, 0)\).
• **Use L'Hôpital's Rule**: Although typically a method in real analysis, a complex version can help in tackling limits of fractions of functions.
• **Focus on Derivatives**: By scrutinizing the derivatives of \(f(z)\) and \(g(z)\), and using respected results like Theorem 5.13, you reduce complex expressions to more manageable forms.

This methodical approach towards division embodies the elegance of complex analysis, allowing what seems to be unsolvable at first glance to become a problem with a neat solution.