# For what constant $c$ does Struve function $\mathbf{H}_1(x) = c$ have infinitely many roots?

I. Sine integral

We start with the plot of the sine integral $$\rm{Si}(x)$$,

The median point $$1.57$$ in fact is $$\frac{\pi}2 \approx 1.5708$$, so the equation $$\rm{Si}(x) = \frac{\pi}2$$ has infinitely many zeros.

II. Struve function

The Struve function $$\mathbf{H}_1(x)$$ has the plot,

Q: What is the exact value of $$c \approx 0.64$$ such that $$\mathbf{H}_1(x) = c$$ has infinitely many zeros? (I don't think it is $$\frac{\pi}5\approx 0.628$$.)

• Your $\frac \pi 5$ was quite good since $\pi^2 \sim 10$. Tomorrow, I shall add an asymptotics to my answer. Cheers :-) – Claude Leibovici Apr 27 at 16:05

$$\lim_{x\to \infty } \, \mathbf{H}_1(x)=\frac 2 \pi \approx 0.63662$$

Have a look here for a good approximation of $$\mathbf{H}_1(x)$$.

Edit

A simple asymptotics of the function for large $$x$$ is

$$\mathbf{H}_1(x)\sim\frac 2 \pi-\frac{(8 x-3) \sin (x)+(8 x+3) \cos (x) } {8\sqrt\pi \,x^{\frac 3 2}}$$ $$\left( \begin{array}{ccc} x & \text{approximation} & \text{exact} \\ 10 & 0.88535469 & 0.89183249 \\ 20 & 0.47115765 & 0.47268818 \\ 30 & 0.72103373 & 0.72175038 \\ 40 & 0.63082698 & 0.63122341 \\ 50 & 0.57982665 & 0.58007845 \\ 60 & 0.72848642 & 0.72866607 \\ 70 & 0.54177739 & 0.54190496 \\ 80 & 0.70601440 & 0.70611511 \\ 90 & 0.61043293 & 0.61051110 \\ 100 & 0.61624769 & 0.61631110 \\ 200 & 0.65192155 & 0.65193751 \\ 300 & 0.66986528 & 0.66987239 \\ 400 & 0.67543349 & 0.67543750 \\ 500 & 0.67073084 & 0.67073340 \\ 600 & 0.65862759 & 0.65862936 \\ 700 & 0.64292911 & 0.64293041 \\ 800 & 0.62773906 & 0.62774005 \\ 900 & 0.61661621 & 0.61661699 \\ 1000 & 0.61183544 & 0.61183608 \end{array} \right)$$

This approximation shows that, for large $$x$$, the successive solutions $$x_k$$ of equation $$\mathbf{H}_1(x)=\frac 2 \pi$$ are "almost" $$x_{k+1}\sim x_k + \pi$$ Computing, for the range of your plot, a few roots rigorously and using the approximation $$\left( \begin{array}{cc} \text{exact} & \text{approximation} \\ 150.008983620 & 150.008549357 \\ 153.149772398 & 153.150193291 \\ 156.292243506 & 156.291835163 \\ 159.433078832 & 159.433475096 \\ 162.575498103 & 162.575113201 \\ 165.716375636 & 165.716749583 \\ 168.858747959 & 168.858384338 \\ 171.999663906 & 172.000017555 \\ 175.141993547 & 175.141649317 \\ 178.282944578 & 178.283279700 \\ 181.425235281 & 181.424908777 \end{array} \right)$$

• Dang, it was just $c = \frac{2}{\pi}$? I thought it would be some other constant, but $\pi$ really is ubiquitous. – Tito Piezas III Apr 27 at 14:20