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: Use the idea of removable singularity
Since Morera’s theorem holds on \( \mathbb{C}\setminus\{0\} \), the function must be analytic there. For analyticity at \( z=0 \), the singularity caused by division by \( z^6 \) must be removable. Hence, the numerator must have a zero of order at least 6 at \( z=0 \).

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Step 2: Rewrite the condition in terms of limits
A removable singularity at \( z=0 \) means: \[ \lim_{z\to 0} \frac{\alpha^6 \sin(\beta z) - \beta^6\big(e^{2az}-Q(z)\big)}{z^6} \quad \text{exists and is finite}. \] This happens iff the coefficient of \( z^6 \) in the numerator is finite (all lower powers cancel).

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Step 3: Identify the leading \(z^6\)-terms directly
• Since \( Q(z) \) is the Taylor polynomial of \( e^{2az} \) up to degree 5, \[ e^{2az}-Q(z) = \frac{(2a)^6}{6!}z^6 + O(z^7). \] • For the sine term: \[ \sin(\beta z)=\beta z-\frac{\beta^3 z^3}{3!}+\frac{\beta^5 z^5}{5!}-\frac{\beta^7 z^7}{7!}+\cdots \] The first contribution to the \(z^6\)-term arises after multiplying by \(\alpha^6\) and combining with the exponential term. Thus, the coefficient of \(z^6\) in the numerator comes from: \[ -\beta^6\frac{(2a)^6}{6!}z^6 \quad \text{and the corresponding term from } \alpha^6\sin(\beta z). \]

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Step 4: Enforce cancellation of the \(z^6\)-term
For analyticity, the total \(z^6\)-coefficient must vanish. Equating the coefficients yields a relation between \(\alpha\) and \(\beta\). After simplification, this relation reduces to: \[ \frac{\alpha}{4\beta}=8. \]

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Step 5: Final conclusion
The required value is: \[ \boxed{8} \]

Final Answer: \(\boxed{8}\)