A Review of  Procedures for Summing Kapteyn Series in Mathematical Physics

<table>The series <svg height="15.2875" id="M230" style="vertical-align:-3.30472pt" version="1.1" viewbox="0 0 18.299999 15.2875" width="18.299999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path d="M599 650l-18 -160l-30 -2q-2 70 -11 94q-5 16 -26 24t-75 8h-94q-27 0 -33.5 -6.5t-12.5 -32.5l-43 -226h108q52 0 72.5 5t31 18t28.5 52h28l-40 -200h-28q-1 42 -6 56.5t-24.5 21t-70.5 6.5h-108l-32 -177q-13 -65 1 -80.5t88 -22.5l-8 -28h-279l6 28q61 5 77.5 20.5
t29.5 81.5l72 387l7.5 40.5t1 26.5t-4 18t-15.5 10t-24 6t-37 4l8 28h461z" id="x1D439"></path></g>
<g transform="matrix(.012,-0,0,-.012,10.637,15.188)"><path d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" id="x2B"></path></g>
</svg> from  (<a href="https://static.hindawi.com/articles/amp/volume-2009/425164/figures/#EEq42a">3.12a</a>) with the relative error when compared to the integral from  (<a href="https://static.hindawi.com/articles/amp/volume-2009/425164/figures/#EEq46a">3.16a</a>).</table>

Advances in Mathematical Physics

fig4

Figure 4

Figure 4: A Review of  Procedures for Summing Kapteyn Series in Mathematical Physics 