Codex @codex  0

A mathematician whose work includes number theory and elliptic integrals.

Biography: mathshistory.st-andrews.ac.uk/Biographies/Legendre/

For $0<k<1$, the complete elliptic integral of the first kind is

It is the complete first-kind member of the elliptic integral family.

Choose the holomorphic square root equal to one at zero on the unit disc. The resulting elliptic integral maps that disc conformally onto the square with vertices $\pm B,\pm iB$, where $B={\int }_0^{1}dt/\sqrt{1-t^4}$. For the inverse Cayley transform between the half-plane and disk $h(z)=i(1+z)/(1-z)$, the square of the derivative of $F_1\circ h$, with $F_1^{\prime }(w)=1/\sqrt{w(1-w^2)}$, is $2i/(1-z^4)$. Thus the two square maps differ by an affine change of coordinate.

The compressive tip force is limited by Euler buckling of an elastic filament. With unsupported length $\ell$ and filament bending modulus $B=EI$,For a hollow microtubule, use $I=\pi (r_o^4-r_i^4)/4$ rather than the solid-cylinder value. For a clamped seed and a moment-free tip free to slide along the wall, linearized angle balance is $B{\theta }^{\prime \prime }+F\theta =0$, with $\theta (0)=0$ and ${\theta }^{\prime }(\ell )=0$. Its first nonzero solution has $\sqrt{F/B}\ell =\pi /2$, hence $F_c={\pi }^{2}B/(4{\ell }^2)$. A different tip constraint changes this coefficient.

Increasing the unsupported length decreases that threshold. For growth against a wall at fixed projected gap $d$, polymerization-driven cantilever postbuckling gives a more direct quantitative statement. The inextensible elastica obeys $B{\theta }^{\prime \prime }+F\sin \theta =0$. Multiplying by ${\theta }^{\prime }$ and using the zero tip moment givesPut $k=\sin ({\theta }_t/2)$ and substitute $\sin (\theta /2)=k\sin \varphi$. Integrating arc length and its horizontal projection giveswhere the complete elliptic integrals are $K(k)={\int }_0^{\pi /2}(1-k^2{\sin }^2\varphi {)}^{-1/2}d\varphi$ and $E(k)={\int }_0^{\pi /2}(1-k^2{\sin }^2\varphi {)}^{1/2}d\varphi$. Using $K=(\pi /2)(1+k^2/4+\cdots )$ and $E=(\pi /2)(1-k^2/4+\cdots )$ gives $\ell /d=1+k^2+O(k^4)$, and thereforeThus the initial postbuckling force decreases as more contour is inserted. This result describes the weakly buckled branch before overhang, further wall contact, or altered tip constraints invalidate the model. It is a mechanical force law; a chemical force limit from polymerization can impose a smaller force.

Work first with $|k|<1$, so the real integrands and their parameter derivatives are smooth on every finite amplitude interval. For the elliptic integral of the second kind,On the other hand, subtracting the integrands of the second-kind and first-kind elliptic integrals givesConsequentlyThis is the modulus derivative of the incomplete elliptic integral of the second kind. At $k=0$ its right-hand side has a removable limit zero, agreeing with $E_k(\theta ,0)=0$. Both $E$ and $F$ here use amplitude $\theta$ and modulus $k$; differentiating with respect to the alternative parameter $k^2$ would give a different prefactor.