Error curves of finite completely monotonic functions associated with inverse power functions by exponential sum approximations

<p>Each figure of page&nbsp;<span class="math-tex">\(M=1,\ldots,17\)</span> in each&nbsp;PDF file represents the error curve&nbsp;of a finite completely monotonic function&nbsp;<span class="math-tex">\(f(x) = \int_a^b \mathrm{e}^{-xt} \frac{t^{\eta - 1}}{\Gamma(\eta)} \mathrm{d}t\)</span>&nbsp; by an <span class="math-tex">\(M\)</span>-term exponential sum approximation. Each PDF file corresponds to the following parameters:</p> <p>inverse_power0.5_a0.5_b1_EMx.pdf for&nbsp;<span class="math-tex">\(\eta = 0.5, a = 2^{-1}, b = 1\)</span><br> inverse_power0.5_a0.0009765625_b1_EMx.pdf for&nbsp;<span class="math-tex">\(\eta = 0.5, a = 2^{-10}, b = 1\)</span><br> inverse_power1_a0.5_b1_EMx.pdf for&nbsp;<span class="math-tex">\(\eta = 1, a = 2^{-1}, b = 1\)</span><br> inverse_power1_a0.0009765625_b1_EMx.pdf for&nbsp;<span class="math-tex">\(\eta = 1, a = 2^{-10}, b = 1\)</span><br> inverse_power2_a0.5_b1_EMx.pdf for&nbsp;<span class="math-tex">\(\eta = 2, a = 2^{-1}, b = 1\)</span><br> inverse_power2_a0.0009765625_b1_EMx.pdf for&nbsp;<span class="math-tex">\(\eta = 2, a = 2^{-10}, b = 1\)</span></p> <p>&nbsp;</p> <p>Each page includes 6 row figures:<br> Row=1 Best exponential sum approximation<br> Row=2 Approximation by the Gaussian quadrature with the variable transformation <span class="math-tex">\(\Phi_{a/b}(u)\)</span><br> Row=3 Approximation by the Gaussian quadrature with the variable transformation <span class="math-tex">\(\varphi_{\exp, a/b}(u)\)</span><br> Row=4 Approximation by the Gaussian quadrature with the variable transformation <span class="math-tex">\(\varphi_{\mathcal{P}_2, a/b}(u)\)</span><br> Row=5 Approximation by the Gaussian quadrature with the variable transformation <span class="math-tex">\(\varphi_{\mathcal{P}_1, a/b}(u)\)</span><br> Row=6 Approximation by the Gaussian quadrature with the variable transformation <span class="math-tex">\(\varphi_{\mathcal{R}_{0, 1}, a/b}(u)\)</span></p> <p>&nbsp;</p> <p>The figures were generated by</p> <p>https://github.com/ymkoyama/fcmf, commit 17342bf3974eee9b4d6b3d21b06f13b87929a85b on January 21, 2023.</p>