{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T06:18:15Z","timestamp":1774678695253,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,12,30]],"date-time":"2019-12-30T00:00:00Z","timestamp":1577664000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We explore a well-known integral representation of the logarithmic function, and demonstrate its usefulness in obtaining compact, easily computable exact formulas for quantities that involve expectations and higher moments of the logarithm of a positive random variable (or the logarithm of a sum of i.i.d. positive random variables). The integral representation of the logarithm is proved useful in a variety of information-theoretic applications, including universal lossless data compression, entropy and differential entropy evaluations, and the calculation of the ergodic capacity of the single-input, multiple-output (SIMO) Gaussian channel with random parameters (known to both transmitter and receiver). This integral representation and its variants are anticipated to serve as a useful tool in additional applications, as a rigorous alternative to the popular (but non-rigorous) replica method (at least in some situations).<\/jats:p>","DOI":"10.3390\/e22010051","type":"journal-article","created":{"date-parts":[[2020,1,3]],"date-time":"2020-01-03T03:28:53Z","timestamp":1578022133000},"page":"51","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["An Integral Representation of the Logarithmic Function with Applications in Information Theory"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9547-3243","authenticated-orcid":false,"given":"Neri","family":"Merhav","sequence":"first","affiliation":[{"name":"The Andrew and Erna Viterbi Faculty of Electrical Engineering, Israel Institute of Technology Technion City, Haifa 3200003, Israel"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5681-1273","authenticated-orcid":false,"given":"Igal","family":"Sason","sequence":"additional","affiliation":[{"name":"The Andrew and Erna Viterbi Faculty of Electrical Engineering, Israel Institute of Technology Technion City, Haifa 3200003, Israel"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"M\u00e9zard, M., and Montanari, A. 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