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Explain why it's well-defined |
→Derivatives with respect to the first variable: differential equations for the amplitude |
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With the [[#Addition theorems|addition theorems above]] and for a given ''m'' with 0 < ''m'' < 1 the major functions are therefore solutions to the following nonlinear [[ordinary differential equation]]s:
* <math>\operatorname{am}(x)</math> solves the differential equations <math>\frac{\mathrm d^2y}{\mathrm dx^2}+m\sin (y)\cos (y)=0</math> and
:<math>\left(\frac{\mathrm dy}{\mathrm dx}\right)^2=1-m\sin(y)^2</math> (for <math>x</math> not on a branch cut)
* <math>\operatorname{sn}(x)</math> solves the differential equations <math>\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1+m) y - 2 m y^3 = 0</math> and <math> \left(\frac{\mathrm{d} y}{\mathrm{d}x}\right)^2 = (1-y^2) (1-m y^2)</math>
* <math>\operatorname{cn}(x)</math> solves the differential equations <math>\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1-2m) y + 2 m y^3 = 0</math> and <math> \left(\frac{\mathrm{d} y}{\mathrm{d}x}\right)^2 = (1-y^2) (1-m + my^2)</math>
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