Problem 4
Question
The function \(f: \mathbb{C} \rightarrow \mathbb{C}\) $$ f(z)= \begin{cases}\exp \left(-1 / z^{4}\right) & \text { for } z \neq 0 \\ 0 & \text { for } z=0\end{cases} $$ satisfies the CAUCHY-RIEMANN equations for all \(z \in \mathbb{C}\) and is complex differentiable for all \(z \in \mathbb{C}^{\bullet}\), but not at the origin.
Step-by-Step Solution
Verified Answer
The function is differentiable for all \(z \neq 0\) but not at the origin due to limit path dependency.
1Step 1: Analyze Function Determination
The given function is \(f(z) = \exp(-1/z^4)\) for \(z eq 0\) and \(f(0) = 0\). This will be defined separately for these two domains.
2Step 2: Check Differentiability at Non-zero Points
For \(z eq 0\), \(f(z) = \exp(-1/z^4)\) is an entire function because \(\exp\) is entire and the composition of entire functions is entire. Therefore, \(f\) is complex differentiable for all \(z eq 0\).
3Step 3: Investigate Behavior at the Origin
At \(z = 0\), we must check the limit of the derivative \(f'(z)\) as \(z\) approaches \(0\). The complex derivative is not defined at \(z = 0\) as the limits leading to the origin in any path yield different results, showing the function is not differentiable at the origin.
4Step 4: Cauchy-Riemann Equations Verification
Since \(f(z)\) is expressed in terms of entire functions (away from the origin), it satisfies the Cauchy-Riemann equations at all \(z eq 0\). For \(z = 0\), these equations don't help because differentiability fails.
Key Concepts
Cauchy-Riemann equationsentire functionscomplex analysis
Cauchy-Riemann equations
The Cauchy-Riemann equations are fundamental in complex analysis, providing a set of two real-valued equations that must be satisfied for a complex function to be differentiable. They can be expressed for a complex function in terms of its real and imaginary parts as follows: if a function is written as \(f(z) = u(x, y) + iv(x, y)\), where \(z = x + iy\), then the Cauchy-Riemann equations are:
However, at points where differentiability fails, such as the origin in our example, these equations cannot be applied effectively to confirm differentiability. Essentially, when the function fails these equations at a point, like the origin here, it shows that the function cannot be complex differentiable at that point.
- \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \)
- \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \)
However, at points where differentiability fails, such as the origin in our example, these equations cannot be applied effectively to confirm differentiability. Essentially, when the function fails these equations at a point, like the origin here, it shows that the function cannot be complex differentiable at that point.
entire functions
Entire functions are an important class in complex analysis. An entire function is one that is complex differentiable everywhere in the complex plane. They are the complex equivalent of differentiable functions in real analysis, but with the broad requirement of differentiability on the entire plane.
Examples include polynomials, the exponential function, and trigonometric functions, all of which fulfill this condition. Our function, \(f(z) = \exp(-1/z^4)\), when \(z eq 0\), is composed essentially of exponential components, known to be entire. The key note, however, is this status is only upheld for \(z eq 0\), as at \(z = 0\), the function becomes non-differentiable. This difference shows that entire functions must remain smooth and consistently differentiable across their entire domain, with any disruption, like at the origin, disqualifying that status for the entire function on the entire plane.
Examples include polynomials, the exponential function, and trigonometric functions, all of which fulfill this condition. Our function, \(f(z) = \exp(-1/z^4)\), when \(z eq 0\), is composed essentially of exponential components, known to be entire. The key note, however, is this status is only upheld for \(z eq 0\), as at \(z = 0\), the function becomes non-differentiable. This difference shows that entire functions must remain smooth and consistently differentiable across their entire domain, with any disruption, like at the origin, disqualifying that status for the entire function on the entire plane.
complex analysis
Complex analysis is a field of mathematics dealing with functions that operate on complex numbers. This vast area explores topics involving analytic functions, the complex equivalent of real analytic functions, which are functions locally represented by convergent power series.
As with our exercise, some key subjects include differentiation, integration, and the applicability of series expansions within the complex domain. A large part of complex analysis is understanding how these functions behave in the complex plane, including criteria for differentiability, such as the Cauchy-Riemann equations.
Potential applications are found in multiple fields, from engineering to physics, where complex functions often model real-world phenomena. Complex analysis provides tools for evaluating behaviors at singularities, like the origin in our exercise, and allows for detailed study of how functions extend across the plane. This exercise, examining differentiability and conditions at non-zero and zero points, encapsulates some of the primary concerns in complex analysis.
As with our exercise, some key subjects include differentiation, integration, and the applicability of series expansions within the complex domain. A large part of complex analysis is understanding how these functions behave in the complex plane, including criteria for differentiability, such as the Cauchy-Riemann equations.
Potential applications are found in multiple fields, from engineering to physics, where complex functions often model real-world phenomena. Complex analysis provides tools for evaluating behaviors at singularities, like the origin in our exercise, and allows for detailed study of how functions extend across the plane. This exercise, examining differentiability and conditions at non-zero and zero points, encapsulates some of the primary concerns in complex analysis.
Other exercises in this chapter
Problem 4
Let \(M \subseteq \mathbb{R}^{p}\). A point \(a \in \mathbb{R}^{p}\) is called an accumulation point of \(M\) if for each \(\varepsilon\)-disk \(U_{\varepsilon}
View solution Problem 4
Let \(f: D \rightarrow \mathbb{C}\) be complex differentiable at \(a \in D\) and \(D^{*}:=\\{z ; \bar{z} \in D\\}\) Then the function \(g: D^{*} \rightarrow \ma
View solution Problem 5
Suppose \(n \in \mathbb{N}\) and \(z_{\nu}, w_{\nu} \in \mathbb{C}\) for \(1 \leq \nu \leq n .\) Prove $$ \left|\sum_{\nu=1}^{n} z_{\nu} w_{\nu}\right|^{2}=\sum
View solution Problem 5
Determine, in each case, all the \(z \in \mathbb{C}\) with $$ \begin{aligned} \exp (z) &=-2, & \exp (z) &=\mathrm{i}, & & \exp (z) &=-\mathrm{i} \\ \sin z &=100
View solution