Suppose f : (−1, 1) → ℝ is an infinitely differentiable function such that the series \(\displaystyle \sum_{j=0}^{\infty} a_j \frac{x^j}{j !}\) converges to f(x) for each x ∈ (−1, 1), where,
\(\rm \displaystyle a_j=\int_\theta^{\pi / 2} \theta^j \cos ^j(\tan \theta) d \theta\) + \(\rm \displaystyle \int_{\pi / 2}^\pi(\theta-\pi)^j \cos ^j(\tan \theta) d \theta\)
for j ≥ 0. Then
1
f(x) = 0 for all x ∈ (-1, 1)
2
f is a non-constant even function on (-1, 1)
3
f is a non-constant odd function on (-1, 1)
4
f is NEITHER an odd function NOR an even function on (-1, 1)