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

Semi-analytic Shape Function of High-harmonic Electron Cyclotron Emission in Tenuous Plasma