https://file.aiquickdraw.com/mathbot/file/17264925582714rpgqy45.jpg

We are asked to show that \(\frac{d}{dz}(\cos z) = -\sin z\), where \(z = x + iy\), with \(x\) and \(y\) being real numbers.

### Step-by-Step Solution:

1. **Express the cosine of a complex number**:
   The complex cosine function can be defined using Euler's formula:
   \[
   \cos z = \frac{e^{iz} + e^{-iz}}{2}
   \]
   where \(z = x + iy\) is a complex number.

2. **Differentiate \(\cos z\) with respect to \(z\)**:
   Using the derivative of exponential functions:
   \[
   \frac{d}{dz} \left( e^{iz} \right) = i e^{iz} \quad \text{and} \quad \frac{d}{dz} \left( e^{-iz} \right) = -i e^{-iz}
   \]
   Therefore, the derivative of \(\cos z\) is:
   \[
   \frac{d}{dz} \left( \cos z \right) = \frac{i e^{iz} - i e^{-iz}}{2}
   \]
   
3. **Simplify the expression**:
   Notice that:
   \[
   \frac{i e^{iz} - i e^{-iz}}{2} = i \left( \frac{e^{iz} - e^{-iz}}{2} \right)
   \]
   From the definition of the sine function:
   \[
   \sin z = \frac{e^{iz} - e^{-iz}}{2i}
   \]
   Therefore, we have:
   \[
   \frac{d}{dz} \left( \cos z \right) = - \sin z
   \]

Thus, we have shown that:
\[
\frac{d}{dz} (\cos z) = -\sin z
\]

### Would you like more details or have any questions? Here are some related problems:

1. Differentiate \(\sin z\) with respect to \(z\) and verify the result.
2. Prove that \(\frac{d}{dz}(e^{iz}) = ie^{iz}\).
3. Show that \(\frac{d}{dz}(\cosh z) = \sinh z\) for a complex variable \(z\).
4. Solve \(\frac{d}{dz} (\sin z) = \cos z\) for \(z = x + iy\).
5. Derive the formula for the derivative of \(\tan z\) with respect to \(z\).

**Tip**: When working with complex functions, Euler's formula is a powerful tool to express trigonometric functions in terms of exponentials, simplifying differentiation and other operations.

Show that \(\frac{d}{dz}(\cos z) = -\sin z\), where \(z = x + iy\).

Explore the detailed proof of the differentiation of the complex cosine function, \(\cos z\), using Euler's formula. The solution demonstrates how to find the derivative \(\frac{d}{dz}(\cos z) = -\sin z\), where \(z = x + iy\). Suitable for undergraduate students studying complex analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation