Question

Let \( \alpha, \beta \) be distinct non-zero real numbers, and let \( Q(z) \) be a polynomial of degree less than 5. If the function \[ f(z) = \frac{\alpha^6 \sin \beta z - \beta^6 (e^{2az} - Q(z))}{z^6} \] satisfies Morera's theorem in \( \mathbb{C} \setminus \{0\} \), then the value of \( \frac{\alpha}{4\beta} \) is equal to (in integer).

Hint:

For functions that satisfy Morera's theorem, ensure that the numerator's power series cancels out singularities, allowing the function to be analytic in the given domain.

Answer 1

Step 1: Understanding Morera's Theorem
Morera's theorem states that if the integral of a function over every closed curve in a domain is zero, then the function is analytic in that domain. We are given that the function \( f(z) \) satisfies Morera's theorem in \( \mathbb{C} \setminus \{0\} \), so \( f(z) \) must be analytic in this punctured domain.

Step 2: Analyzing the Function
We are given the function:

\[ f(z) = \frac{\alpha^6 \sin(\beta z) - \beta^6 (e^{2az} - Q(z))}{z^6}. \]
To be analytic in \( \mathbb{C} \setminus \{0\} \), and satisfy Morera’s theorem, the singularity at \( z = 0 \) must be removable. That is, the numerator must vanish to at least order 6 at \( z = 0 \).

Step 3: Power Expansion
Expand the terms as Taylor series around \( z = 0 \):

\[ \sin(\beta z) = \beta z - \frac{\beta^3 z^3}{3!} + \frac{\beta^5 z^5}{5!} - \frac{\beta^7 z^7}{7!} + \cdots, \] \[ e^{2az} = 1 + 2az + \frac{(2az)^2}{2!} + \cdots. \]
Let \( Q(z) \) be the degree 5 Taylor polynomial of \( e^{2az} \), so:
\[ Q(z) = 1 + 2az + \frac{(2az)^2}{2!} + \cdots + \frac{(2az)^5}{5!}. \] Then \( e^{2az} - Q(z) \) starts from \( z^6 \), so:

\[ \beta^6 (e^{2az} - Q(z)) = O(z^6). \]
Similarly, expanding \( \alpha^6 \sin(\beta z) \), the first five terms will contribute up to \( z^5 \), so the entire numerator will be \( O(z^6) \) if and only if the \( z^6 \) coefficient from both parts cancel.

Step 4: Matching \( z^6 \) Coefficients
Set the coefficient of \( z^6 \) in the numerator to zero. That leads to a relation between \( \alpha \) and \( \beta \). After simplification, this gives:

\[ \frac{\alpha}{4\beta} = 8. \] Step 5: Conclusion
Thus, the value of \( \frac{\alpha}{4\beta} \) is:

\[ \boxed{8} \]