Let's analyze whether the function \( f(z) = e^{2x} (\cos 2y + i \sin 2y) \) is analytic.

### Step 1: Express the function in terms of \( z = x + iy \)
Given:

\[
f(z) = e^{2x} (\cos 2y + i \sin 2y)
\]

Recall that the complex exponential function can be expressed as:

\[
e^{i\theta} = \cos \theta + i \sin \theta
\]

This implies:

\[
\cos 2y + i \sin 2y = e^{i(2y)}
\]

Thus, the function can be rewritten as:

\[
f(z) = e^{2x} \cdot e^{i2y} = e^{2(x + iy)} = e^{2z}
\]

### Step 2: Separate \( f(z) \) into real and imaginary parts
Let \( z = x + iy \). Then:

\[
f(z) = e^{2z} = e^{2(x + iy)} = e^{2x} \cdot e^{i2y}
\]

Expanding this:

\[
f(z) = e^{2x} (\cos 2y + i \sin 2y)
\]

The real part \( u(x, y) \) and imaginary part \( v(x, y) \) of the function are:

\[
u(x, y) = e^{2x} \cos 2y, \quad v(x, y) = e^{2x} \sin 2y
\]

### Step 3: Apply the Cauchy-Riemann equations
The Cauchy-Riemann equations are:

\[
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
\]

Let's compute the partial derivatives.

#### For \( u(x, y) = e^{2x} \cos 2y \):

\[
\frac{\partial u}{\partial x} = 2e^{2x} \cos 2y, \quad \frac{\partial u}{\partial y} = -2e^{2x} \sin 2y
\]

#### For \( v(x, y) = e^{2x} \sin 2y \):

\[
\frac{\partial v}{\partial x} = 2e^{2x} \sin 2y, \quad \frac{\partial v}{\partial y} = 2e^{2x} \cos 2y
\]

### Step 4: Check if the Cauchy-Riemann equations hold

- \(\frac{\partial u}{\partial x} = 2e^{2x} \cos 2y\) and \(\frac{\partial v}{\partial y} = 2e^{2x} \cos 2y\): These are equal.
- \(\frac{\partial u}{\partial y} = -2e^{2x} \sin 2y\) and \(\frac{\partial v}{\partial x} = 2e^{2x} \sin 2y\): These are also equal when multiplied by \(-1\).

Since both Cauchy-Riemann equations are satisfied everywhere in the complex plane, the function \( f(z) = e^{2z} \) is **analytic**.

Would you like more details or have any further questions?

### Related Questions:
1. What are the properties of the function \( e^{z} \) in the complex plane?
2. How do exponential functions behave in the complex plane?
3. What is the significance of the Cauchy-Riemann equations in complex analysis?
4. Can you explain how the Cauchy-Riemann equations ensure the analyticity of a function?
5. What are some examples of non-analytic functions and their properties?

### Tip:
In complex analysis, functions like \( e^{z} \) are commonly encountered and are known to be analytic throughout the entire complex plane, which makes them holomorphic (analytic) everywhere.

Explore the analyticity of the function f(z) = e^{2x} (cos 2y + i sin 2y) through detailed analysis. Understand the application of complex exponential functions and Cauchy-Riemann equations. Suitable for college-level students and those interested in complex analysis.

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