Find and classify the singularities of f. Show all steps: f(z)= sin(1/z)cos(1/z)

We are tasked with finding and classifying the singularities of the function \( f(z) = \sin\left(\frac{1}{z}\right) \cos\left(\frac{1}{z}\right) \). Let’s go through the steps one by one.

### Step 1: Analyze the function

We start by noting that the given function can be rewritten using a trigonometric identity. Recall the identity:

\[
\sin(A) \cos(A) = \frac{1}{2} \sin(2A)
\]

Applying this to \( A = \frac{1}{z} \), we get:

\[
f(z) = \sin\left(\frac{1}{z}\right) \cos\left(\frac{1}{z}\right) = \frac{1}{2} \sin\left(\frac{2}{z}\right)
\]

Thus, the function becomes:

\[
f(z) = \frac{1}{2} \sin\left(\frac{2}{z}\right)
\]

### Step 2: Identify the singularities

To identify the singularities of \( f(z) \), we look for points where the function is not analytic. In this case, the function \( \sin\left(\frac{2}{z}\right) \) involves \( \frac{2}{z} \), which becomes problematic when \( z = 0 \), as division by zero occurs.

Thus, the point \( z = 0 \) is a singularity of the function. No other singularities exist because the sine function \( \sin\left(\frac{2}{z}\right) \) is analytic everywhere else in the complex plane, except at \( z = 0 \).

### Step 3: Classify the singularity at \( z = 0 \)

Now we need to classify the singularity at \( z = 0 \). Singularities can generally be classified as one of the following:
1. **Removable singularity**
2. **Pole**
3. **Essential singularity**

To classify the singularity at \( z = 0 \), we investigate the behavior of \( f(z) \) as \( z \to 0 \).

#### Step 3.1: Behavior of \( f(z) \) as \( z \to 0 \)

The function \( f(z) = \frac{1}{2} \sin\left(\frac{2}{z}\right) \) oscillates wildly as \( z \to 0 \). The sine function \( \sin\left(\frac{2}{z}\right) \) has no limit as \( z \) approaches 0; instead, it exhibits an essential singularity at \( z = 0 \) because the function does not tend to infinity or a finite limit, and instead oscillates between all possible values in an unpredictable way near \( z = 0 \).

#### Step 3.2: Laurent series near \( z = 0 \)

We can further confirm that \( z = 0 \) is an essential singularity by examining the Laurent series for \( \sin\left(\frac{2}{z}\right) \). The Laurent series expansion of \( \sin\left(\frac{1}{z}\right) \) around \( z = 0 \) involves an infinite number of negative powers of \( z \), which is a characteristic of an essential singularity.

Thus, \( z = 0 \) is an **essential singularity**.

### Conclusion

The function \( f(z) = \sin\left(\frac{1}{z}\right) \cos\left(\frac{1}{z}\right) \) has a singularity at \( z = 0 \), and this singularity is classified as an **essential singularity**.

---

Would you like a detailed explanation of any of the steps or a clarification on singularity classifications? Here are five related questions you might find helpful:

1. How do you compute the Laurent series of a function around a singularity?
2. What is the difference between an essential singularity and a pole?
3. Can you classify singularities using the residue of a function?
4. How does the behavior of \( \sin\left(\frac{1}{z}\right) \) as \( z \to 0 \) confirm an essential singularity?
5. What are examples of other functions with essential singularities?

**Tip:** A function with an essential singularity can take on almost every complex value infinitely often near that point (Picard's Theorem).

Learn how to find and classify the singularities of the function f(z) = sin(1/z)cos(1/z) with a detailed step-by-step explanation. This problem covers complex analysis concepts such as essential singularities and Laurent series expansions. Suitable for undergraduate students.

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