Problem 40
Question
We have \(f^{\prime}(z)=\lim _{\Delta z \rightarrow 0} \frac{(z+\Delta z)(\bar{z}+\overline{\Delta z})-z \bar{z}}{\Delta z}=\lim _{\Delta z \rightarrow 0}\left(\bar{z}+z \frac{\overline{\Delta z}}{\Delta z}+\overline{\Delta z}\right).\) If \(z=0,\) then the above limit becomes $$f^{\prime}(0)=\lim _{\Delta z \rightarrow 0} \overline{\Delta z}=0.$$ If \(z \neq 0\) then we first let \(\Delta z \rightarrow 0\) along a horizontal line so that \(\Delta z=\Delta x\) and \(\overline{\Delta z}=\Delta x .\) Thus, $$f^{\prime}(z)=\lim _{\Delta z \rightarrow 0}\left(\bar{z}+z \frac{\Delta x}{\Delta x}+\Delta x\right)=\bar{z}+z.$$ Next we let \(\Delta z \rightarrow 0\) along a vertical line so that \(\Delta z=i \Delta y, \overline{\Delta z}=-i \Delta y .\) Thus $$f^{\prime}(z)=\lim _{\Delta y \rightarrow 0}\left(\bar{z}+z \frac{-i \Delta y}{i \Delta y}+i \Delta y\right)=\bar{z}-z.$$ We must have \(\bar{z}+z=\bar{z}-z\) which implies \(z=0\).This is a contradiction to the assumption that \(z \neq 0 .\) Hence \(f(z)=|z|^{2}\) is differentiable only at \(z=0.\)
Step-by-Step Solution
Verified Answer
The function \( f(z) = |z|^2 \) is differentiable only at \( z = 0 \).
1Step 1: Understand the Expression
The given expression for \[ f^{\prime}(z)=\lim _{\Delta z \rightarrow 0} \frac{(z+\Delta z)(\bar{z}+\overline{\Delta z})-z \bar{z}}{\Delta z}=\lim _{\Delta z \rightarrow 0}\left(\bar{z}+z \frac{\overline{\Delta z}}{\Delta z}+\overline{\Delta z}\right) \]is the derivative of a function involving complex numbers. Our goal is to evaluate this expression under different conditions.
2Step 2: Evaluate when \( z = 0 \)
When \( z = 0 \), we substitute into the expression and simplify: \[ f^{\prime}(0)=\lim _{\Delta z \rightarrow 0} \overline{\Delta z}=0. \] This shows that the function \( f(z) \) is differentiable at \( z = 0 \), yielding a derivative of zero.
3Step 3: Evaluate for Horizontal Approach when \( z \neq 0 \)
For \( z eq 0 \), consider \( \Delta z \rightarrow 0 \) along a horizontal line (i.e., \( \Delta z = \Delta x \), thus \( \overline{\Delta z} = \Delta x \)). The expression becomes: \[ f^{\prime}(z)=\lim _{\Delta z \rightarrow 0}\left(\bar{z}+z \frac{\Delta x}{\Delta x}+\Delta x\right)=\bar{z}+z. \] This shows a derivative of \( \bar{z} + z \) for this path.
4Step 4: Evaluate for Vertical Approach when \( z \neq 0 \)
Consider \( \Delta z \rightarrow 0 \) along a vertical line (i.e., \( \Delta z = i \Delta y \), thus \( \overline{\Delta z} = -i \Delta y \)). The expression becomes:\[ f^{\prime}(z)=\lim _{\Delta y \rightarrow 0}\left(\bar{z}+z \frac{-i \Delta y}{i \Delta y}+i \Delta y\right)=\bar{z}-z. \] This shows a derivative of \( \bar{z} - z \) for this path.
5Step 5: Identify the Contradiction
For the function to be differentiable, the derivative needs to be path-independent. Both horizontal and vertical evaluations must match for differentiability. However, \( \bar{z} + z = \bar{z} - z \) leads to a contradiction unless \( z = 0 \).
6Step 6: Conclusion: Determining Differentiability
Since the only solution to the path-independent requirement \( \bar{z} + z = \bar{z} - z \) is when \( z = 0 \), the function \( f(z) = |z|^2 \) is differentiable only at \( z = 0 \).
Key Concepts
Complex NumbersPath IndependenceComplex AnalysisDifferentiability of Complex Functions
Complex Numbers
Complex numbers are numbers composed of a real part and an imaginary part. They are usually expressed in the form \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \).
The introduction of complex numbers allows us to solve certain equations that have no solution in the realm of real numbers alone. In our given problem, these numbers are denoted by the variable \( z \) and involve their complex conjugates. The complex conjugate of \( z = a + bi \) is \( \bar{z} = a - bi \).
Complex conjugation is useful in simplifying expressions and solving equations involving complex functions. These operations with complex numbers, involving their conjugates, play a vital role in evaluating differentiability in the complex plane.
The introduction of complex numbers allows us to solve certain equations that have no solution in the realm of real numbers alone. In our given problem, these numbers are denoted by the variable \( z \) and involve their complex conjugates. The complex conjugate of \( z = a + bi \) is \( \bar{z} = a - bi \).
Complex conjugation is useful in simplifying expressions and solving equations involving complex functions. These operations with complex numbers, involving their conjugates, play a vital role in evaluating differentiability in the complex plane.
Path Independence
Path independence is an important concept in complex analysis and refers to the idea that a function's derivative or integral should yield the same result, irrespective of the path taken in the complex plane.
In the exercise, to evaluate differentiability, the derivative was computed along two paths: horizontally, where \( \Delta z = \Delta x \) and \( \overline{\Delta z} = \Delta x \), and vertically, where \( \Delta z = i \Delta y \) and \( \overline{\Delta z} = -i \Delta y \). The results from these different paths showed disagreement, an indication of path dependence, which implies that the function is not differentiable except at the point \( z = 0 \).
Ensuring path independence is crucial because it verifies the well-definedness and consistency of mathematical manipulations involving complex derivatives and integrals.
In the exercise, to evaluate differentiability, the derivative was computed along two paths: horizontally, where \( \Delta z = \Delta x \) and \( \overline{\Delta z} = \Delta x \), and vertically, where \( \Delta z = i \Delta y \) and \( \overline{\Delta z} = -i \Delta y \). The results from these different paths showed disagreement, an indication of path dependence, which implies that the function is not differentiable except at the point \( z = 0 \).
Ensuring path independence is crucial because it verifies the well-definedness and consistency of mathematical manipulations involving complex derivatives and integrals.
Complex Analysis
Complex analysis is the study of functions of a complex variable. It extends real analysis into the complex domain and provides a rich theory encompassing differentiation, integration, series, and residue calculus.
This field helps in understanding various properties of complex functions and their applications. One of the key tools in complex analysis is evaluating differentiability in terms of the Cauchy-Riemann equations, which provide necessary and sufficient conditions for differentiability of complex-valued functions.
In our scenario, we used the foundational principles of complex analysis to explore where the function \( f(z) = |z|^2 \) is differentiable. Such exploration often involves calculating limits and understanding the implications of path independence. This investigation informs us about the behavior of the function under different conditions and paths.
This field helps in understanding various properties of complex functions and their applications. One of the key tools in complex analysis is evaluating differentiability in terms of the Cauchy-Riemann equations, which provide necessary and sufficient conditions for differentiability of complex-valued functions.
In our scenario, we used the foundational principles of complex analysis to explore where the function \( f(z) = |z|^2 \) is differentiable. Such exploration often involves calculating limits and understanding the implications of path independence. This investigation informs us about the behavior of the function under different conditions and paths.
Differentiability of Complex Functions
Differentiability in the complex plane is more stringent than in the real realm. For a complex function \( f(z) \) to be differentiable at a point \( z \), the derivative must exist and be the same irrespective of the approach direction.
In the problem, \( f(z) \' = \, \lim_{\Delta z \rightarrow 0} \frac{(z + \Delta z)(\bar{z} + \overline{\Delta z}) - z\bar{z}}{\Delta z} \) was analyzed by approaching horizontally and vertically. These approaches yielded different results unless \( z = 0 \), indicating the function is only differentiable at that point.
Hence, differentiability of complex functions is largely about path independence and meeting special conditions in the complex plane. In this sense, complex differentiation powerfully captures the essence of function behavior within and across the complex universe.
In the problem, \( f(z) \' = \, \lim_{\Delta z \rightarrow 0} \frac{(z + \Delta z)(\bar{z} + \overline{\Delta z}) - z\bar{z}}{\Delta z} \) was analyzed by approaching horizontally and vertically. These approaches yielded different results unless \( z = 0 \), indicating the function is only differentiable at that point.
Hence, differentiability of complex functions is largely about path independence and meeting special conditions in the complex plane. In this sense, complex differentiation powerfully captures the essence of function behavior within and across the complex universe.
Other exercises in this chapter
Problem 39
We have $$\lim _{\Delta z \rightarrow 0} \frac{\overline{z+\Delta z}-\bar{z}}{\Delta z}=\lim _{\Delta z \rightarrow 0} \frac{\overline{\Delta z}}{\Delta z}.$$ I
View solution Problem 39
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View solution Problem 40
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View solution Problem 41
Each linear equation in the system \(\frac{d x}{d t}=2 x, \quad \frac{d y}{d t}=2 y\) can be solved directly. We obtain \(x(t)=c_{1} e^{2 t}\) and \(y(t)=c_{2}
View solution