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Magazine Chapter 27: Problem 9
Define the function
$$
f_{N}(z)=\cot z-\sum_{n=-N}^{N} \frac{1}{z-\pi n}
$$
and show that \(f(z)=\lim _{N \rightarrow \infty} f_{N}(z)\) vanishes everywhere
in the complex plane. Hints: (a) Show that \(f\) is antisymmetric, \(f(-z)=-f(z)
.\) (b) Show that \(f(z)\) is periodic with period \(\Delta x=\pi .\) (c) Show that
\(f(z)\) is holomorphic in the strip \(-\pi / 2
Short Answer
Expert verified
The function \( f(z) \) vanishes everywhere in the complex plane.
Step by step solution
01
Prove Antisymmetry of f
To show that the function is antisymmetric, we calculate \( f_{N}(-z) \) and simplify: \[ f_{N}(-z) = \cot(-z) - \sum_{n=-N}^{N} \frac{1}{-z-\pi n} = -\cot(z) + \sum_{n=-N}^{N} \frac{1}{z+\pi n}. \]We can relabel the sum's index \( k = -n \), which flips the range, and due to properties of the cotangent and symmetry of the sums,\[ f(-z) = -f_(z). \]Thus, \( f \) is antisymmetric: \( f(-z) = -f(z) \).
02
Establish Periodicity of f
To show periodicity, calculate \( f_{N}(z + \pi) \):\[ f_{N}(z + \pi) = \cot(z+\pi) - \sum_{n=-N}^{N} \frac{1}{z + \pi - \pi n}. \]Using \( \cot(z+\pi) = \cot(z) \), it follows that:\[ f_{N}(z + \pi) = \cot(z) - \sum_{n=-N}^{N} \frac{1}{z - \pi n} = f_{N}(z). \]Therefore, \( f(z) \) is periodic with period \( \pi \).
03
Show Holomorphy in the Strip
In the strip \(-\pi/2 < x < \pi/2\), \( \, \cot(z) \) is holomorphic - it has no poles as it is not integer multiples of \( \pi \). The constructed function can be written without singularities as the terms cancel pole contributions inside the strip. Thus, \( f(z) \) is holomorphic in this strip.
04
Verify Vanishing on Boundaries
On the boundaries of the strip \( x = \pm \pi/2 \), the function needs to vanish. Evaluating at these points, as \( N \to \infty \), each term in the sum approximates and cancels the \( \cot(z) \), resulting in\[ \lim_{N \to \infty} f_{N} \left( \pm \frac{\pi}{2} + iy \right) = 0. \]So, \( f(z) \to 0 \) on the strip's boundaries.
05
Conclusion Using a General Theorem
Since \( f(z) \) is holomorphic in the strip and vanishes on the boundaries, it vanishes identically in the strip according to the theorem. By periodicity, \( f(z) \) vanishes in every strip of width \( \pi \), leading to \( f(z) = 0 \) everywhere in the complex plane.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Holomorphic Functions
In the realm of complex analysis, a holomorphic function is a complex-valued function that is differentiable at every point in its domain. What's unique about holomorphic functions is that they are not just differentiable in the real sense; their differentiability extends to complex numbers. This means that small changes in the complex plane lead to smooth transformations of the function.
- Holomorphic functions are infinitely differentiable.
- They obey Cauchy-Riemann equations, which provide a condition for a function to be differentiable in the complex plane.
- In the context of our exercise, the function \( f(z) \) is holomorphic in the strip \(-\pi/2 < x < \pi/2\) because it has no poles or singularities within this region.
Antisymmetric Functions
An antisymmetric function is essentially a function that, when evaluated at \(-z\), returns the negative of the value at \(z\). Mathematically, this is expressed as \( f(-z) = -f(z) \). This property is fascinating because it gives the function a type of mirror-image behavior across the y-axis, when dealing with real components.
- If you swap any two input variables, the output is negated.
- This symmetry means that about certain axes or points, the function鈥檚 value abruptly changes signs.
- In our specific exercise, this antisymmetric nature of \( f \) was established by showing \( f_{N}(-z) \) equates to \(-f_{N}(z) \). This showcases a perfect reflection or cancellation in waveforms, linking closely to symmetries often seen in physics and signal processing.
Periodicity in Complex Functions
The concept of periodicity indicates that a function repeats its values in a regular interval. In complex functions, periodicity helps in analyzing patterns and behavior over infinite domains. If a function is periodic with period \(T\), this means:
- For any value of \( z \), \( f(z + T) = f(z) \).
- In simpler terms, the function behaves predictably after every interval of \(T\).
- Periodic functions simplify complex analysis problems since they allow repeated sections of the domain to be studied independently.
- This periodicity, combined with holomorphic and antisymmetric properties, contributes to the eventual conclusion that the function vanishes throughout the complex plane.
Boundary Value Theorem
The boundary value theorem is a vital principle in complex analysis, especially when dealing with functions defined over certain regions like strips or disks. This theorem states that if a function is holomorphic and it vanishes on the boundary of a region, it must also vanish everywhere in that region.
- This theorem is a powerful tool because it bridges local behavior at boundaries with global behavior throughout the region.
- In our exercise, applying this theorem shows that since \( f(z) \) vanishes on the boundary of the strip \(-\pi/2 < x < \pi/2\), it must also vanish everywhere within the strip.
