# Heaviside's theory of signal transmission on submarine cables

Banach Center Publications (2010)

- Volume: 88, Issue: 1, page 159-173
- ISSN: 0137-6934

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topHikosaburo Komatsu. "Heaviside's theory of signal transmission on submarine cables." Banach Center Publications 88.1 (2010): 159-173. <http://eudml.org/doc/281781>.

@article{HikosaburoKomatsu2010,

abstract = {As written in L. Schwartz' book, Heaviside's theory of cables is an important source of the theory of generalized functions. The partial differential equations he discussed were the usual heat equation and the simplest hyperbolic equations of one space dimension, but he had to solve them as evolution equations in the unusual direction of the distance along which the electric signals propagate. Although he obtained explicit expressions of solutions, which were of great economical values, it has not yet been clarified completely how he derived and proved them. The author gives easy proofs of Heaviside's results based only on the classical theory of Laplace transforms and their reciprocal, the Bromwich integrals. At the end it is indicated that an abstract version of the Fatou theorem on bounded harmonic functions on a half space implies the uniqueness of solutions for the Thomson cables.},

author = {Hikosaburo Komatsu},

journal = {Banach Center Publications},

keywords = {Thomson's theory of cables; Heaviside's operational calculus; generalised functions; Laplace transforms; Bromwich integrals},

language = {eng},

number = {1},

pages = {159-173},

title = {Heaviside's theory of signal transmission on submarine cables},

url = {http://eudml.org/doc/281781},

volume = {88},

year = {2010},

}

TY - JOUR

AU - Hikosaburo Komatsu

TI - Heaviside's theory of signal transmission on submarine cables

JO - Banach Center Publications

PY - 2010

VL - 88

IS - 1

SP - 159

EP - 173

AB - As written in L. Schwartz' book, Heaviside's theory of cables is an important source of the theory of generalized functions. The partial differential equations he discussed were the usual heat equation and the simplest hyperbolic equations of one space dimension, but he had to solve them as evolution equations in the unusual direction of the distance along which the electric signals propagate. Although he obtained explicit expressions of solutions, which were of great economical values, it has not yet been clarified completely how he derived and proved them. The author gives easy proofs of Heaviside's results based only on the classical theory of Laplace transforms and their reciprocal, the Bromwich integrals. At the end it is indicated that an abstract version of the Fatou theorem on bounded harmonic functions on a half space implies the uniqueness of solutions for the Thomson cables.

LA - eng

KW - Thomson's theory of cables; Heaviside's operational calculus; generalised functions; Laplace transforms; Bromwich integrals

UR - http://eudml.org/doc/281781

ER -