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Say I have an incomplete elliptic integral of first kind of the form

$$F(\varphi(z), k(z))=\int_0^{\varphi(z)} \frac{d \theta}{\sqrt{1-k(z)^2 \sin ^2 \theta}}$$

where each arguments are function of some variable $z$.

How do I taylor series $F$ in small z?

Thanks!

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  • $\begingroup$ Each of the derivatives $$\frac{\partial^{a+b}}{\partial^a \varphi \,\partial^b k}\,F(\varphi,k)$$can be computed, but it will be complicated to write them in a single formula. $\endgroup$
    – Gerald Edgar
    Commented May 11 at 12:09
  • $\begingroup$ Where can I find the formula for the derivatives? $\endgroup$
    – Quantization
    Commented May 11 at 14:10

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