How Long is The Graph of xⁿ?

How long is the graph of xⁿ? Logarithmic corner defects in arc length. Zishaan Ahmed (independent researcher), 2026. The graph of y = xⁿ on [0,1] has arc length L(n) → 2 as n → ∞. This paper determines how fast: 2 − L(n) = (ln n + 1 − ln 2)/(n − 1) + O(ln²n / n²) L(n) = 1 + (2/(en))^{1/(n−1)} + π²/(12n²) + O(ln n / n³) an exact, rapidly convergent Gamma-function formula for L(n), obtained from a single Beta integral corners of any turning angle θ: the coefficient of (ln n)/n is 2 sin²(θ/2) by contrast, the squircle |x|ᵖ + |y|ᵖ = 1 has perimeter 8 − 4K/p + O(1/p²) with K = 2√2 l n(1+√2) − 2 ln 2, with no logarithm