Question

Consider the complex function \( f(z) = \cos z + e^{z^2 \). The coefficient of \( z^5 \) in the Taylor series expansion of \( f(z) \) about the origin is \_\_\_\_\_ (rounded off to 1 decimal place).}

Hint:

The Taylor series expansion of a function about the origin involves terms with only integer powers of \( z \). Check each term carefully.

Answer 1

Given: \[ f(z) = \cos z + e^{z^2} \] Step 1: Series expansion of \(\cos z\): \[ \cos z = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \frac{z^6}{6!} + \cdots \] Step 2: Series expansion of \(e^{z^2}\): \[ e^{z^2} = 1 + \frac{z^2}{1!} + \frac{z^4}{2!} + \frac{z^6}{3!} + \cdots \] Step 3: Combining the expansions: \[ f(z) = \left(1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \cdots \right) + \left(1 + \frac{z^2}{1!} + \frac{z^4}{2!} + \cdots \right) \] Step 4: Coefficient of \( z^5 \): From the series expansion, there is no term involving \( z^5 \). Final Answer: The coefficient of \( z^5 \) is \( 0.0 \). % Quick tip