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:<math>M(z)=z\exp\left(-\int_0^z\int_0^w \left(\frac{1}{\operatorname{sl}^2v}-\frac{1}{v^2}\right)\, \mathrm dv\,\mathrm dw\right),</math>
:<math>N(z)=\exp\left(\int_0^z\int_0^w \operatorname{sl}^2v\,\mathrm dv\,\mathrm dw\right)</math>
where the contours do not cross the poles; while the innermost integrals are path-independent, the outermost ones are path-dependent; however, the path dependence cancels out with the non-injectivity of the complex [[exponential function]].
An alternative way of expressing the lemniscate functions as a ratio of entire functions involves the theta functions (see [[#Methods of computation|Lemniscate elliptic functions § Methods of computation]]); the relation between <math>M,N</math> and <math>\theta_1,\theta_3</math> is
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=== Use in integration ===
The lemniscate arcsine can be used to integrate many functions. Here is a list of important integrals (the [[Constant of integration|constants of integration]] are omitted):
:<math>\int\frac{1}{\sqrt{1-x^4}}\,\mathrm dx=\operatorname{arcsl} x</math>
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:<math> \operatorname{slh}(a+b) = \frac{\operatorname{slh}a\operatorname{slh}'b + \operatorname{slh}b\operatorname{slh}'a}{1-\operatorname{slh}^2a\,\operatorname{slh}^2b} </math>
When <math>u</math> is real, the derivative and the original [[antiderivative]] of <math> \operatorname{slh} </math> and <math> \operatorname{clh} </math> can be expressed in this way:
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