Derivative of $f(x)=\int_1^{\frac{1}{x}}\frac{\text{d}t}{\sqrt{(t^2-1)(1-t^2x^2)}}$

Question

This function:

$$f(x)=\int_1^{\frac{1}{x}}\frac{\text{d}t}{\sqrt{(t^2-1)(1-t^2x^2)}}$$ where $x\in (0,1)$

I want to find its derivative.

In order to solve this question. I go back to general cases: $$f(x)=\int_a^{\varphi(x)}F(x,t)\text{d}t$$ By definition: $$f'(x)=\lim_{\Delta x \rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$

$$=\lim_{\Delta x \rightarrow 0}\frac{\int_a^{\varphi(x+\Delta x)}F(x+\Delta x,t)\text{d}t-\int_a^{\varphi(x)}F(x,t)\text{d}t}{\Delta x}=$$ $$\lim_{\Delta x \rightarrow 0}\frac{1}{\Delta x}\int_a^{\varphi(x+\Delta x)}[F(x+\Delta x,t)-F(x,t)]\text{d}t + \lim_{\Delta x \rightarrow 0}\frac{1}{\Delta x} \int_{\varphi(x)}^{\varphi(x+\Delta x)}F(x,t)\text{d}t=$$ $$\int_a^{\varphi(x)}\frac{\partial F}{\partial x}(x,t)\text{d}t+\varphi'(x)F(x,\varphi(x))$$

But I use this to the function mentioned in the beginning. there are something wrong:

because $F(x,\varphi(x))$ is meaningless.

Does this function have derivatives ? How to find it?

Do you mean for the integrand to be complex-valued? (Because it is for values of $t<1$). — JPi, May 26 '14 at 03:09
See the Leibniz integral rule about differentiating under the integral sign. — Lucian, May 26 '14 at 03:40
In order for the elliptic integral to be real-valued, the elliptic modulus $x$ needs to be restricted $|x|<1$. So the function you've defined will only be real-valued on the domain where both $x$ and $\frac{1}{x}$ have magnitude less than one. You see the problem? — David H, May 26 '14 at 03:46
@JPi, sorry .I made a mistake. the integral is from $1$,not $0$ — Laura, May 26 '14 at 03:47
@DavidH I have edited the post. the integral lower bound is $1$, and $x \in (0,1)$. then the integral is real-valued. — Laura, May 26 '14 at 03:54
@ReneSchipperus because $\varphi(x)=\frac{1}{x}$. then $F(x,\varphi(x))=\frac{1}{\sqrt{(1-\frac{1}{x^2})(1-1)}}$ so it is meaningless — Laura, May 26 '14 at 03:56

score 2 · Answer 1 · answered May 26 '14 at 13:56

2

It is an elliptic integral of the first kind. The derivative is expressed in terms of elliptic integrals of the first and second kind.

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answered May 26 '14 at 13:56

JJacquelin

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Derivative of $f(x)=\int_1^{\frac{1}{x}}\frac{\text{d}t}{\sqrt{(t^2-1)(1-t^2x^2)}}$

1 Answers1