There are solutions at E = ± i cosh − 1 ⁡ ( 1 / e ) , {\displaystyle E=\pm i\cosh ^{-1}(1/e),} and at those values M = E − e sin ⁡ E = ± i ( cosh − 1 ⁡ ( 1 / e ) − 1 − e 2 ) {\displaystyle M=E-e\sin E=\pm i\left(\cosh ^{-1}(1/e)-{\sqrt {1-e^{2}}}\right)} (where inverse cosh is taken to be positive), and dE/dM goes to infinity at these points.
在 E = ± i cosh − 1 ⁡ ( 1 / e ) {\displaystyle E=\pm i\cosh ^{-1}(1/e)} 有解，而在這些值時 M = E − e sin ⁡ E = ± i ( cosh − 1 ⁡ ( 1 / e ) − 1 − e 2 ) {\displaystyle M=E-e\sin E=\pm i\left(\cosh ^{-1}(1/e)-{\sqrt {1-e^{2}}}\right)} （其中逆cosh取正值），而dE/dM在這些點則趨向無限。