Mathieu Functions and Hill’s Equation - 28.22 Connection Formulas

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
28.22.E5	${\displaystyle{\displaystyle g_{\mathit{e},2m}(h)=(-1)^{m}\sqrt{\dfrac{2}{\pi}}\dfrac{\mathrm{ce}_{2m}\left(\frac{1}{2}\pi,h^{2}\right)}{A_{0}^{2m}(h^{2})}}}$ g_{\mathit{e},2m}(h) = (-1)^{m}\sqrt{\dfrac{2}{\pi}}\dfrac{\Mathieuce{2m}@{\frac{1}{2}\pi}{h^{2}}}{A_{0}^{2m}(h^{2})}		g[e , 2*m](h) = (- 1)^(m)*sqrt((2)/(Pi))*(MathieuCE(2*m, (h)^(2), (1)/(2)*Pi))/((A[0])^(2*m)((h)^(2)))	Subscript[g, e , 2*m][h] == (- 1)^(m)*Sqrt[Divide[2,Pi]]*Divide[MathieuC[2*m, (h)^(2), Divide[1,2]*Pi],(Subscript[A, 0])^(2*m)[(h)^(2)]]	Failure	Failure	Error	Failed [300 / 300] Result: Plus[Complex[0.5000000000000001, 0.8660254037844386], Times[Complex[0.42295231653869036, 0.41961671574834936], Power[A, -1]]] Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 1], Rule[Subscript[A, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, E, Times[2, m]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.5000000000000001, 0.8660254037844386], Times[Complex[-0.38839890891671613, -0.3454183210952864], Power[A, -1]]] Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 2], Rule[Subscript[A, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, E, Times[2, m]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
28.22.E6	${\displaystyle{\displaystyle g_{\mathit{e},2m+1}(h)=(-1)^{m+1}\sqrt{\frac{2}{\pi}}\dfrac{\mathrm{ce}_{2m+1}'\left(\frac{1}{2}\pi,h^{2}\right)}{hA_{1}^{2m+1}(h^{2})}}}$ g_{\mathit{e},2m+1}(h) = (-1)^{m+1}\sqrt{\frac{2}{\pi}}\dfrac{\Mathieuce{2m+1}'@{\frac{1}{2}\pi}{h^{2}}}{hA_{1}^{2m+1}(h^{2})}		g[e , 2*m + 1](h) = (- 1)^(m + 1)*sqrt((2)/(Pi))*(subs( temp=(1)/(2)*Pi, diff( MathieuCE(2*m + 1, (h)^(2), temp), temp$(1) ) ))/((hA[1])^(2*m + 1)((h)^(2)))	Subscript[g, e , 2*m + 1][h] == (- 1)^(m + 1)*Sqrt[Divide[2,Pi]]*Divide[D[MathieuC[2*m + 1, (h)^(2), temp], {temp, 1}]/.temp-> Divide[1,2]*Pi,(Subscript[hA, 1])^(2*m + 1)[(h)^(2)]]	Failure	Failure	Error	Failed [300 / 300] Result: Plus[Complex[0.5000000000000001, 0.8660254037844386], Times[Complex[-0.2975776534545682, -0.6256781760348913], Power[A, -1]]] Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 1], Rule[Subscript[A, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, E, Plus[1, Times[2, m]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.5000000000000001, 0.8660254037844386], Times[Complex[-0.42963849355864525, 0.8495253193240367], Power[A, -1]]] Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 2], Rule[Subscript[A, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, E, Plus[1, Times[2, m]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
28.22.E7	${\displaystyle{\displaystyle g_{\mathit{o},2m+1}(h)=(-1)^{m}\sqrt{\dfrac{2}{\pi}}\dfrac{\mathrm{se}_{2m+1}\left(\frac{1}{2}\pi,h^{2}\right)}{hB_{1}^{2m+1}(h^{2})}}}$ g_{\mathit{o},2m+1}(h) = (-1)^{m}\sqrt{\dfrac{2}{\pi}}\dfrac{\Mathieuse{2m+1}@{\frac{1}{2}\pi}{h^{2}}}{hB_{1}^{2m+1}(h^{2})}		g[o , 2*m + 1](h) = (- 1)^(m)*sqrt((2)/(Pi))*(MathieuSE(2*m + 1, (h)^(2), (1)/(2)*Pi))/((hB[1])^(2*m + 1)((h)^(2)))	Subscript[g, o , 2*m + 1][h] == (- 1)^(m)*Sqrt[Divide[2,Pi]]*Divide[MathieuS[2*m + 1, (h)^(2), Divide[1,2]*Pi],(Subscript[hB, 1])^(2*m + 1)[(h)^(2)]]	Failure	Failure	Error	Failed [300 / 300] Result: Plus[Complex[0.5000000000000001, 0.8660254037844386], Times[Complex[-0.32036211571699924, -0.11607109445443671], Power[B, -1]]] Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 1], Rule[o, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, o, Plus[1, Times[2, m]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.5000000000000001, 0.8660254037844386], Times[Complex[-0.1322357993555902, 0.30696697344841817], Power[B, -1]]] Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 2], Rule[o, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, o, Plus[1, Times[2, m]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
28.22.E8	${\displaystyle{\displaystyle g_{\mathit{o},2m+2}(h)=(-1)^{m+1}\sqrt{\dfrac{2}{\pi}}\dfrac{\mathrm{se}_{2m+2}'\left(\frac{1}{2}\pi,h^{2}\right)}{h^{2}B_{2}^{2m+2}(h^{2})}}}$ g_{\mathit{o},2m+2}(h) = (-1)^{m+1}\sqrt{\dfrac{2}{\pi}}\dfrac{\Mathieuse{2m+2}'@{\frac{1}{2}\pi}{h^{2}}}{h^{2}B_{2}^{2m+2}(h^{2})}		g[o , 2*m + 2](h) = (- 1)^(m + 1)*sqrt((2)/(Pi))*(subs( temp=(1)/(2)*Pi, diff( MathieuSE(2*m + 2, (h)^(2), temp), temp$(1) ) ))/((h)^(2)* (B[2])^(2*m + 2)((h)^(2)))	Subscript[g, o , 2*m + 2][h] == (- 1)^(m + 1)*Sqrt[Divide[2,Pi]]*Divide[D[MathieuS[2*m + 2, (h)^(2), temp], {temp, 1}]/.temp-> Divide[1,2]*Pi,(h)^(2)* (Subscript[B, 2])^(2*m + 2)[(h)^(2)]]	Failure	Failure	Error	Failed [300 / 300] Result: Plus[Complex[0.5000000000000001, 0.8660254037844386], Times[Complex[0.09053953879094334, 2.773543957850464], Power[B, -1]]] Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 1], Rule[o, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, o, Plus[2, Times[2, m]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.5000000000000001, 0.8660254037844386], Times[Complex[-0.7797636104550828, -1.7837750479423518], Power[B, -1]]] Test Values: {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[m, 2], Rule[o, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, o, Plus[2, Times[2, m]]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data