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The Euler formula, sometimes also called the Euler identity (e.g., Trott 2004, p. 174), states e^(ix)=cosx+isinx, (1) where i is the imaginary unit. Note that Euler's ...

The Helmholtz differential equation is not separable in toroidal coordinates

Euler integration was defined by Schanuel and subsequently explored by Rota, Chen, and Klain. The Euler integral of a function f:R->R (assumed to be piecewise-constant with ...

The third-order ordinary differential equation y^(''')+alphayy^('')+beta(1-y^('2))=0.

The Helmholtz differential equation is not separable in bispherical coordinates.

Whittaker and Watson (1990, pp. 539-540) write Lamé's differential equation for ellipsoidal harmonics of the first kind of the four types as ...

The ordinary differential equation y^('')+k/xy^'+deltae^y=0.

In bipolar coordinates, the Helmholtz differential equation is not separable, but Laplace's equation is.

The second-order ordinary differential equation y^('')-[(m(m+1)+1/4-(m+1/2)cosx)/(sin^2x)+(lambda+1/2)]y=0.

The second-order ordinary differential equation y^('')+[alpha/(cosh^2(ax))+betatanh(ax)+gamma]y=0.