  Semi-analytic expressions for the electron cyclotron emission (ECE) shape
function are developed for arbitrary high harmonics. The integrand of the
${m}^{\mathrm{th}}$ harmonic ECE shape function is fitted with the
readily-integrable test function which is parameterized by plasma temperature
$T_e$, harmonic number $m$ and emission angle $\theta $. Semi-analytic formulae
for high harmonic ECE emissivity are obtained by integrating the test integrand
with the fitting parameters gained from the regression analysis as well as the
inductive studies for arbitrary cases. The developed expression matches the
numerically-integrated ECE shape function very well and the overall differences
between original shape functions and semi-analytic shape functions are
evaluated. The expression can be used for rapid analysis of high-harmonic ECE
spectra.
