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for {{nowrap|''k<sub>i</sub>'' {{=}} 0, 1, ..., ''N<sub>i</sub>'' − 1}}, ''i'' = 1, 2, ..., ''r''.
== Multidimensional Laplace
The multidimensional Laplace transform is useful for the solution of boundary value problems. Boundary value problems in two or more variables characterized by partial differential equations can be solved by a direct use of the Laplace transform.<ref name=":0">{{Cite journal|title = Theorems on multidimensional laplace transform for solution of boundary value problems|url = http://www.sciencedirect.com/science/article/pii/089812218990031X|journal = Computers & Mathematics with Applications|date = 1989-01-01|pages = 1033–1056|volume = 18|issue = 12|doi = 10.1016/0898-1221(89)90031-X|first = Joyati|last = Debnath|first2 = R. S.|last2 = Dahiya}}</ref> The Laplace transform for an M-dimensional case is defined<ref name=":0"/> as
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F(x,y) is called the image of f(x,y) and f(x,y) is known as the original of F(x,y).<ref name=":1">{{Cite journal|url = |title = Multi-Dimensional Laplace Transforms and Systems of Partial Differential Equations |last = Aghili and Moghaddam|first = |journal = International Mathematical Forum |volume=1 |year=2006 |issue=21 |pages=1043–1050|doi = |pmid = |access-date = }}</ref> This special case can be used to solve the [[Telegrapher's equations]].<ref name=":1" />
== Multidimensional Z
The multidimensional Z transform is used to map the discrete time ___domain multidimensional signal to the Z ___domain. This can be used to check the stability of filters. The equation of the multidimensional Z transform is given by
[[File:Figure 1.1a depicting region of support.png|thumb|209x209px|Figure 1.1a]]
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<math display="inline"> z=e^{jw} </math> where z and w are vectors.
=== Region of
[[File:Region of Convergence for figure 1.1a.png|thumb|201x201px|Figure 1.1b]]
Points (z1,z2) for which <math>F(z_1,z_2)=\sum_{n_1=-\infty}^\infty \sum_{n_2=-\infty}^\infty |f(n_1,n_2)| |z_1|^{-n_1} |z_2|^{-n_2}</math> <math><\infty</math> are located in the ROC.
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