Fig. 5. The fundamental contour ${\mathscr L}$ for straight line edges (shown left). The spectral rays $\lbrace {\mathscr L}_j \,|\, j=1,2,3 \rbrace $

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Fig. 5. The fundamental contour ${\mathscr L}$ for straight line edges (shown left). The spectral rays $\lbrace {\mathscr L}_j \,|\, j=1,2,3 \rbrace $ from Fokas & Kapaev (2003) for the $N=3$ polygon $P$ shown in Fig. 4 (left). The latter rays are images of ${\mathscr L}$ under the $N$ transformations (2.14). From: A transform method for Laplace's equation in multiply connected circular domains IMA J Appl Math. 2015;80(6):1902-1931. doi:10.1093/imamat/hxv019 IMA J Appl Math | © The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

Fig. 4. A convex polygon $P$ as an intersection of $N=3$ half-planes with $N$ angles $\lbrace \chi _j \,|\, j=1,2,3 \rbrace $. Formula (2.7) can be used in the Cauchy integral formula with $\chi =\chi _j$ when $z'$ is on side $S_j$ (for $j=1,2,3$). From: A transform method for Laplace's equation in multiply connected circular domains IMA J Appl Math. 2015;80(6):1902-1931. doi:10.1093/imamat/hxv019 IMA J Appl Math | © The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

Fig. 3. Geometrical positioning of $z$ and $z'$ for the validity of (2 From: A transform method for Laplace's equation in multiply connected circular domains IMA J Appl Math. 2015;80(6):1902-1931. doi:10.1093/imamat/hxv019 IMA J Appl Math | © The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

Fig. 1. The ‘disc-in-channel’ geometry $D$ is a doubly connected circular domain. From: A transform method for Laplace's equation in multiply connected circular domains IMA J Appl Math. 2015;80(6):1902-1931. doi:10.1093/imamat/hxv019 IMA J Appl Math | © The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

Fig. 6. The fundamental contour for circular edges. From: A transform method for Laplace's equation in multiply connected circular domains IMA J Appl Math. 2015;80(6):1902-1931. doi:10.1093/imamat/hxv019 IMA J Appl Math | © The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

Fig. 7. Geometry of $z'$ and $z$ for derivation of (3.6). From: A transform method for Laplace's equation in multiply connected circular domains IMA J Appl Math. 2015;80(6):1902-1931. doi:10.1093/imamat/hxv019 IMA J Appl Math | © The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

Fig. 2. Geometrical positioning of $z$ and $z'$ for the validity of (2 From: A transform method for Laplace's equation in multiply connected circular domains IMA J Appl Math. 2015;80(6):1902-1931. doi:10.1093/imamat/hxv019 IMA J Appl Math | © The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

Fig. 10. Uniform flow with unit speed parallel to the $x$-axis past two equal unit radius cylinders with centres at $y=\pm h$ (left). By the symmetry with respect to the $x$-axis, the problem can be studied in the domain $D$ shown right. From: A transform method for Laplace's equation in multiply connected circular domains IMA J Appl Math. 2015;80(6):1902-1931. doi:10.1093/imamat/hxv019 IMA J Appl Math | © The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

Fig. 8. A multiply connected circular domain $D$ (for $M=2$). From: A transform method for Laplace's equation in multiply connected circular domains IMA J Appl Math. 2015;80(6):1902-1931. doi:10.1093/imamat/hxv019 IMA J Appl Math | © The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

Fig. 9. A doubly connected polygonal domain $P$ Fig. 9. A doubly connected polygonal domain $P$. There is no single transform expression valid uniformly in $P$; however, it can be subdivided into the four convex polygonal subregions shown. From: A transform method for Laplace's equation in multiply connected circular domains IMA J Appl Math. 2015;80(6):1902-1931. doi:10.1093/imamat/hxv019 IMA J Appl Math | © The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

Fig. 12. Graph of $\lambda /(2a)$ against $a$ as computed by the transform method (solid lines) and as given by the approximate formula $\lambda _{\rm approx}$. The crosses show the numerical data computed by Teo & Khoo (2010). The dashed line shows the dilute limit found by Crowdy (2011). From: A transform method for Laplace's equation in multiply connected circular domains IMA J Appl Math. 2015;80(6):1902-1931. doi:10.1093/imamat/hxv019 IMA J Appl Math | © The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

Fig. 11. Longitudinal flow $w(x,y)$, into the page and for $x >0$, past a periodic array of semi-circular bubbles centred at $2 n l{\rm i},\ n \in \mathbb {Z}$. The flow takes place in $x >0$ but the problem can be extended to the whole channel region exterior to a circular inclusion. From: A transform method for Laplace's equation in multiply connected circular domains IMA J Appl Math. 2015;80(6):1902-1931. doi:10.1093/imamat/hxv019 IMA J Appl Math | © The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

Fig. 13. Other common circular domain geometries to which the new transform scheme applies. From: A transform method for Laplace's equation in multiply connected circular domains IMA J Appl Math. 2015;80(6):1902-1931. doi:10.1093/imamat/hxv019 IMA J Appl Math | © The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.