The question in the uploaded image asks to find the residue at $z = 0$ of the function $f(z) = \frac{1+e^z}{\sin z + z \cos z}$.

Step-by-Step Solution:

The residue at $z = 0$ can be computed by examining the Laurent series of $f(z)$ around $z = 0$. Alternatively, since $z = 0$ is likely a pole (as the denominator becomes 0 there), the residue can be calculated using standard residue formulas.

1. Expand the denominator $\sin z + z \cos z$: Recall the Taylor series expansions:

   • $\sin z = z - \frac{z^3}{6} + \frac{z^5}{120} + \dots$,
   • $\cos z = 1 - \frac{z^2}{2} + \frac{z^4}{24} + \dots$.

   So: $\sin z + z \cos z = z + z(1 - \frac{z^2}{2} + \frac{z^4}{24}) = z + z - \frac{z^3}{2} + \frac{z^5}{24} = 2z - \frac{z^3}{2} + \dots$

2. Expand the numerator $1 + e^z$: Recall the Taylor series for $e^z$: $e^z = 1 + z + \frac{z^2}{2} + \frac{z^3}{6} + \dots$ So: $1 + e^z = 1 + (1 + z + \frac{z^2}{2} + \frac{z^3}{6}) = 2 + z + \frac{z^2}{2} + \frac{z^3}{6} + \dots$

3. Simplify $f(z)$: Using the series expansions: $f(z) = \frac{1 + e^z}{\sin z + z \cos z} = \frac{2 + z + \frac{z^2}{2} + \dots}{2z - \frac{z^3}{2} + \dots}.$

   Divide the series term by term: $f(z) = \frac{2}{2z} + \frac{z}{2z} + \frac{\frac{z^2}{2}}{2z} + \dots = \frac{1}{z} + \frac{1}{2} + \frac{z}{4} + \dots$

4. Identify the residue: The residue at $z = 0$ is the coefficient of $\frac{1}{z}$ in the Laurent series expansion. From the simplification above, the residue is: $\text{Residue} = 1.$

Final Answer:

The residue of $f(z) = \frac{1+e^z}{\sin z + z \cos z}$ at $z = 0$ is 1.

Would you like further clarification or additional steps?

Related Questions:

1. How do you find residues for higher-order poles?
2. What is the residue theorem and how is it applied in contour integration?
3. How does the Laurent series differ from the Taylor series?
4. Can you explain how the expansion of the denominator influences the result?
5. What are the common applications of residue theory in complex analysis?

Tip:

When expanding functions in series, always check the lowest-order term of the denominator to determine the nature of the singularity (pole or essential).