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### 1. **Conformal Mapping**
You are asked to prove that the transformation \( w = \frac{2 + iz}{4z + i} \) maps the real axis in the \( z \)-plane into a circle in the \( w \)-plane. You also need to find the center and radius of the circle.

- A **conformal mapping** preserves angles and locally behaves like a scaling transformation.
- Begin by substituting \( z = x \), where \( x \in \mathbb{R} \) (since \( z \) is real).
- The result will give you a parametrized form of a circle in the \( w \)-plane. Once you simplify, you'll find the equation of the circle and can determine its center and radius from there.

### 2. **Cauchy Integral Formula**
You need to apply the **Cauchy integral formula** to evaluate:
\[
\int_C \frac{1}{e^{(z^2 - 1)^2}(z + 2)} dz,
\]
where \( C \) is a counterclockwise circle \( |z - 1| = 3 \).

- The **Cauchy Integral Formula** states:
\[
f(a) = \frac{1}{2\pi i} \int_C \frac{f(z)}{z - a} dz
\]
for a function \( f \) analytic inside and on a positively oriented contour \( C \).
- The formula might require some algebraic manipulation of the integrand to apply it.

### 3. **Laurent's Theorem**
You are required to expand the function
\[
f(z) = \frac{1}{(z - 1)(z - 5)}
\]
at \( z = 4 \), using the **Taylor series** expansion, and find the **region of convergence**.

- A **Laurent series** is a representation of a complex function as a power series, which may include terms with negative powers.
- You should expand the given function around \( z = 4 \) and determine in which region the expansion converges (for example, in terms of radii of convergence from the singularities).

### 4. **Cauchy Residue Theorem**
Here, you are asked to use the **Cauchy Residue Theorem** to compute the contour integral:
\[
\int_0^\pi \frac{d\theta}{2 - 3\sin(\theta)}.
\]

- The **Residue Theorem** states that for a meromorphic function, the contour integral around a closed curve \( C \) is \( 2\pi i \times \) the sum of residues of the function inside \( C \).
- Convert the integral into a complex contour integral by using the substitution \( z = e^{i\theta} \). The sine function \( \sin(\theta) \) can be expressed as:
  \[
  \sin(\theta) = \frac{z - z^{-1}}{2i}.
  \]
  Then, apply the Residue Theorem to solve the contour integral.

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### Would you like detailed solutions for any specific problem? Let me know which you'd like me to expand on!

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Here are five questions that extend the above content:
1. How does one compute the center and radius of a circle in the complex plane from a given transformation?
2. What conditions must a function satisfy to apply the Cauchy Integral Formula?
3. How do you determine the region of convergence for a Laurent series in different domains?
4. What is the connection between real integrals and contour integrals in the complex plane?
5. How does the residue at a pole of a function affect the contour integral?

**Tip**: Always look for symmetries in complex integrals—they can simplify contour integration dramatically!

The user uploaded an image containing four complex analysis problems involving conformal mapping, Cauchy integral formula, Laurent's theorem, and the Cauchy residue theorem.

Explore detailed solutions to complex analysis problems involving conformal mapping, the Cauchy integral formula, Laurent's theorem, and the Cauchy residue theorem. This content covers advanced mathematical concepts suitable for undergraduate students studying complex functions.

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