Math Problem Statement
For the function f(z) = cos(z), if z ≠ ±π/2, and f(z) = -1/π, if z = ±π/2, which of the following is true? A. z = ±π/2 are singular points B. z = ±π/2 are simple poles C. z = ±π/2 are removable singularities D. f(z) is an entire function
Solution
Let's analyze the given function:
\begin{cases} \cos z & \text{if } z \neq \pm \frac{\pi}{2}, \\ -\frac{1}{\pi} & \text{if } z = \pm \frac{\pi}{2}. \end{cases}$$ ### Step-by-step reasoning: - The function $$f(z)$$ is defined piecewise. At $$z \neq \pm \frac{\pi}{2}$$, it behaves as $$\cos z$$, which is a well-known entire function (holomorphic over the entire complex plane). - At $$z = \pm \frac{\pi}{2}$$, we have a special definition, setting $$f(z) = -\frac{1}{\pi}$$, different from what $$\cos z$$ would naturally take at these points (since $$\cos z$$ has singularities at these points, where $$z = \pm \frac{\pi}{2}$$). Let's consider the behavior at $$z = \pm \frac{\pi}{2}$$: - The function $$\cos z$$ has simple poles at $$z = \pm \frac{\pi}{2}$$, which suggests singularities. - However, the function $$f(z)$$ redefines the values at these points to $$-\frac{1}{\pi}$$, potentially "removing" the singularities. #### Removable singularities We can check if these points $$z = \pm \frac{\pi}{2}$$ are removable singularities. For removable singularities, if the limit of the function as $$z \to \pm \frac{\pi}{2}$$ exists and matches the value of the function at that point, the singularity is removable. Let's compute the limit as $$z \to \pm \frac{\pi}{2}$$: $$\lim_{z \to \pm \frac{\pi}{2}} \left( \cos z \cdot \left( z^2 - \left( \frac{\pi}{2} \right)^2 \right)^2 \right) = 0$$ Thus, the limit exists, and the function is made continuous by setting $$f(z) = -\frac{1}{\pi}$$. Therefore, $$z = \pm \frac{\pi}{2}$$ are removable singularities. ### Answer: The correct statement is **C: $$z = \pm \frac{\pi}{2}$$ are removable singularities**. Would you like more details on the singularities or any further explanation? Here are 5 related questions: 1. What is the definition of a removable singularity? 2. How are singularities classified in complex analysis? 3. What are simple poles in the context of complex functions? 4. Can you explain the behavior of the function $$\cos z$$ near its singularities? 5. How does the concept of "entire function" differ from other types of holomorphic functions? **Tip:** Always check the behavior of limits around singularities to determine their type—whether they are poles, essential, or removable.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Complex Analysis
Singularities
Holomorphic Functions
Formulas
Limit of f(z) as z approaches ±π/2
cos(z) function
Theorems
Removable singularity theorem
Pole classification
Suitable Grade Level
University/College level
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