{"id":48884,"date":"2026-07-13T11:16:54","date_gmt":"2026-07-13T04:16:54","guid":{"rendered":"https:\/\/mathed.bru.ac.th\/?p=48884"},"modified":"2026-07-13T11:16:54","modified_gmt":"2026-07-13T04:16:54","slug":"%e0%b8%ad%e0%b8%b5%e0%b8%a7%e0%b8%b2%e0%b8%99-%e0%b8%9b%e0%b8%a3%e0%b8%b5%e0%b8%a7%e0%b8%b2%e0%b8%a5%e0%b8%ad%e0%b8%9f-%e0%b8%99%e0%b8%b1%e0%b8%81%e0%b8%84%e0%b8%93%e0%b8%b4%e0%b8%95%e0%b8%a8","status":"publish","type":"post","link":"https:\/\/mathed.bru.ac.th\/en\/2026\/07\/13\/%e0%b8%ad%e0%b8%b5%e0%b8%a7%e0%b8%b2%e0%b8%99-%e0%b8%9b%e0%b8%a3%e0%b8%b5%e0%b8%a7%e0%b8%b2%e0%b8%a5%e0%b8%ad%e0%b8%9f-%e0%b8%99%e0%b8%b1%e0%b8%81%e0%b8%84%e0%b8%93%e0%b8%b4%e0%b8%95%e0%b8%a8\/","title":{"rendered":"Ivan Privalov: The mathematician who unlocked the secrets of the peripheral behavior of analytic functions."},"content":{"rendered":"

Ivan Privalov: The mathematician who unlocked the secrets of the peripheral behavior of analytic functions.<\/h1>\n

Lucin's students in the seminar.<\/h2>\n

Ivan Ivanovich Privalov was born on February 11, 1891, in Nizhny Lomov, Penza Oblast, Russia. He graduated from Moscow University in 1913 under the tutelage of Dmitri Egorov and was a contemporary of Nikolai Luzin, who was seven years older but studying at the same time. Privalov was an enthusiastic member of the student mathematics club and served as its president before Egorov, impressed by his abilities, recommended that he stay at Moscow University to pursue research.<\/p>\n

After completing his master's degree in 1916, Privalov became a professor at Saratov University in 1917 before returning to Moscow in 1922 and remaining a professor of complex variable function theory for the rest of his life.<\/p>\n

The marginal behavior of analytic functions.<\/h2>\n

Privalov's early research in 1914 concerned orthogonal series and integral equations, before he moved on to study the properties of Fourier series. However, his main interest that made his name memorable was... The marginal behavior of analytic functions.<\/strong> (Boundary properties of analytic functions) This refers to the study of how complex functions behave as they approach the boundaries of their defined region, particularly near singular points.<\/p>\n

One of his most important works is known as... Plemelj\u2013Privalov Theorem<\/strong> Named jointly with the Slovenian mathematician Josip Plemelj, this theorem describes the invariance of H\u00f6lder classes under the Cauchy singular integral operator. It is a key tool in analytic function theory and has applications in boundary value problems found in both physics and engineering.<\/p>\n

Another theorem bearing his name is: Luzin\u2013Privalov Theorem<\/strong> It was developed in collaboration with Nikolai Luzhin, his teacher, classmate, and mentor.<\/p>\n

From Fourier series to the subharmonic function.<\/h2>\n

In 1934, Privalov turned to academic study. Subharmonic function<\/strong> (Subharmonic functions) This concept was first introduced in 1906 and became a major topic in Riesz's work between 1925 and 1930 in his book. Subharmonic Functions<\/em> (Published in 1937) Privalov presented a classical systemic structure of this type of functional theory, closely linked to harmonic functions theory. Another of his important works is... Boundary Properties of Single-Valued Analytic Functions<\/em> Published in 1941, it was described by critics of the time as encompassing one of the most difficult and nuanced branches of complex variable theory.<\/p>\n

\u0e2b\u0e19\u0e31\u0e07\u0e2a\u0e37\u0e2d Introduction to the Theory of Functions of a Complex Variable<\/em> His work (first published in 1948) has been reprinted numerous times and remains a key reference text in complex analysis courses at many universities to this day.<\/p>\n

A strict but influential teacher.<\/h2>\n

Privalov was renowned as an extremely meticulous teacher, with contemporary memoirs stating that not everyone could tolerate his strictness. As a result, he had fewer students compared to other teachers, but his profound knowledge was widely recognized. He was elected an associate member of the USSR Academy of Sciences in 1939 and served as vice president of the Moscow Mathematical Society from 1936.<\/p>\n

Privalov died in Moscow on July 13, 1941.<\/p>\n

Thought-provoking activities<\/h2>\n
    \n
  1. Why is the behavior of analytic functions \"near the bounds\" more challenging than studying their behavior within regions where the analytic function is well-defined?<\/li>\n
  2. Try researching the meanings of harmonic and subharmonic functions and explain the difference in your own words.<\/li>\n
  3. The Plemelj\u2013Privalov theorem relates to the Cauchy singular integral operator. Let's explore some of its applications in engineering or physics.<\/li>\n
  4. Why did a highly strict teacher like Privalov often have fewer students, yet at the same time command such high respect? What did the students think of this teaching style?<\/li>\n<\/ol>\n

    References<\/h2>\n
      \n
    • O’Connor, J. J., & Robertson, E. F. “Ivan Ivanovich Privalov.” MacTutor History of Mathematics Archive<\/em>, University of St Andrews.<\/li>\n
    • Wikipedia. “Ivan Privalov.”<\/li>\n
    • Encyclopedia.com. “Privalov, Ivan Ivanovich.”<\/li>\n
    • Kuznetsov, P. I., & Solomentsev, E. D. (1982). Ivan Ivanovich Privalov (ninety years after his birth). Russian Mathematical Surveys<\/em>, 37, 152\u2013174.<\/li>\n<\/ul>\n
      \n

      This article is part of the \"Today in the History of Mathematics\" series.<\/em><\/p>\n

      \"\"

      \u0e2d\u0e35\u0e27\u0e32\u0e19 \u0e1b\u0e23\u0e35\u0e27\u0e32\u0e25\u0e2d\u0e1f: \u0e19\u0e31\u0e01\u0e04\u0e13\u0e34\u0e15\u0e28\u0e32\u0e2a\u0e15\u0e23\u0e4c\u0e1c\u0e39\u0e49\u0e44\u0e02\u0e04\u0e27\u0e32\u0e21\u0e25\u0e31\u0e1a\u0e1e\u0e24\u0e15\u0e34\u0e01\u0e23\u0e23\u0e21\u0e0a\u0e32\u0e22\u0e02\u0e2d\u0e1a\u0e02\u0e2d\u0e07\u0e1f\u0e31\u0e07\u0e01\u0e4c\u0e0a\u0e31\u0e19\u0e27\u0e34\u0e40\u0e04\u0e23\u0e32\u0e30\u0e2b\u0e4c \u0e25\u0e39\u0e01\u0e28\u0e34\u0e29\u0e22\u0e4c\u0e43\u0e19\u0e27\u0e07\u0e2a\u0e31\u0e21\u0e21\u0e19\u0e32\u0e02\u0e2d\u0e07\u0e25\u0e39\u0e0b\u0e34\u0e19 \u0e2d\u0e35\u0e27\u0e32\u0e19 \u0e2d\u0e35\u0e27\u0e32\u0e42\u0e19\u0e27\u0e34\u0e0a \u0e1b\u0e23\u0e35\u0e27\u0e32\u0e25\u0e2d\u0e1f (Ivan Ivanovich Privalov) \u0e40\u0e01\u0e34\u0e14\u0e40\u0e21\u0e37\u0e48\u0e2d\u0e27\u0e31\u0e19\u0e17\u0e35\u0e48 11 \u0e01\u0e38\u0e21\u0e20\u0e32\u0e1e\u0e31\u0e19\u0e18\u0e4c \u0e04.\u0e28. 1891 \u0e17\u0e35\u0e48\u0e40\u0e21\u0e37\u0e2d\u0e07\u0e19\u0e34\u0e08\u0e19\u0e35 \u0e42\u0e25\u0e21\u0e2d\u0e1f \u0e41\u0e04\u0e27\u0e49\u0e19\u0e40\u0e1e\u0e19\u0e0b\u0e32 \u0e1b\u0e23\u0e30\u0e40\u0e17\u0e28\u0e23\u0e31\u0e2a\u0e40\u0e0b\u0e35\u0e22 \u0e40\u0e02\u0e32\u0e2a\u0e33\u0e40\u0e23\u0e47\u0e08\u0e01\u0e32\u0e23\u0e28\u0e36\u0e01\u0e29\u0e32\u0e08\u0e32\u0e01\u0e21\u0e2b\u0e32\u0e27\u0e34\u0e17\u0e22\u0e32\u0e25\u0e31\u0e22\u0e21\u0e2d\u0e2a\u0e42\u0e01\u0e43\u0e19\u0e1b\u0e35 \u0e04.\u0e28. 1913 \u0e20\u0e32\u0e22\u0e43\u0e15\u0e49\u0e01\u0e32\u0e23\u0e14\u0e39\u0e41\u0e25\u0e02\u0e2d\u0e07\u0e14\u0e21\u0e34\u0e17\u0e23\u0e35 \u0e40\u0e22\u0e01\u0e2d\u0e23\u0e2d\u0e1f (Dmitri Egorov) \u0e41\u0e25\u0e30\u0e23\u0e48\u0e27\u0e21\u0e2a\u0e21\u0e31\u0e22\u0e01\u0e31\u0e1a\u0e19\u0e34\u0e42\u0e04\u0e44\u0e25 \u0e25\u0e39\u0e0b\u0e34\u0e19 (Nikolai Luzin) \u0e1c\u0e39\u0e49\u0e2d\u0e32\u0e22\u0e38\u0e21\u0e32\u0e01\u0e01\u0e27\u0e48\u0e32\u0e40\u0e02\u0e32\u0e40\u0e08\u0e47\u0e14\u0e1b\u0e35\u0e41\u0e15\u0e48\u0e01\u0e33\u0e25\u0e31\u0e07\u0e28\u0e36\u0e01\u0e29\u0e32\u0e2d\u0e22\u0e39\u0e48\u0e43\u0e19\u0e0a\u0e48\u0e27\u0e07\u0e40\u0e27\u0e25\u0e32\u0e40\u0e14\u0e35\u0e22\u0e27\u0e01\u0e31\u0e19 \u0e1b\u0e23\u0e35\u0e27\u0e32\u0e25\u0e2d\u0e1f\u0e40\u0e1b\u0e47\u0e19\u0e2a\u0e21\u0e32\u0e0a\u0e34\u0e01\u0e17\u0e35\u0e48\u0e01\u0e23\u0e30\u0e15\u0e37\u0e2d\u0e23\u0e37\u0e2d\u0e23\u0e49\u0e19\u0e02\u0e2d\u0e07\u0e0a\u0e21\u0e23\u0e21\u0e04\u0e13\u0e34\u0e15\u0e28\u0e32\u0e2a\u0e15\u0e23\u0e4c\u0e19\u0e31\u0e01\u0e28\u0e36\u0e01\u0e29\u0e32 \u0e41\u0e25\u0e30\u0e14\u0e33\u0e23\u0e07\u0e15\u0e33\u0e41\u0e2b\u0e19\u0e48\u0e07\u0e1b\u0e23\u0e30\u0e18\u0e32\u0e19\u0e0a\u0e21\u0e23\u0e21 \u0e01\u0e48\u0e2d\u0e19\u0e17\u0e35\u0e48\u0e40\u0e22\u0e01\u0e2d\u0e23\u0e2d\u0e1f\u0e08\u0e30\u0e1b\u0e23\u0e30\u0e17\u0e31\u0e1a\u0e43\u0e08\u0e43\u0e19\u0e04\u0e27\u0e32\u0e21\u0e2a\u0e32\u0e21\u0e32\u0e23\u0e16\u0e02\u0e2d\u0e07\u0e40\u0e02\u0e32\u0e08\u0e19\u0e41\u0e19\u0e30\u0e19\u0e33\u0e43\u0e2b\u0e49\u0e2d\u0e22\u0e39\u0e48\u0e15\u0e48\u0e2d\u0e17\u0e35\u0e48\u0e21\u0e2b\u0e32\u0e27\u0e34\u0e17\u0e22\u0e32\u0e25\u0e31\u0e22\u0e21\u0e2d\u0e2a\u0e42\u0e01\u0e40\u0e1e\u0e37\u0e48\u0e2d\u0e17\u0e33\u0e27\u0e34\u0e08\u0e31\u0e22 \u0e2b\u0e25\u0e31\u0e07\u0e2a\u0e33\u0e40\u0e23\u0e47\u0e08\u0e1b\u0e23\u0e34\u0e0d\u0e0d\u0e32\u0e42\u0e17\u0e43\u0e19\u0e1b\u0e35 \u0e04.\u0e28. 1916 \u0e1b\u0e23\u0e35\u0e27\u0e32\u0e25\u0e2d\u0e1f\u0e44\u0e14\u0e49\u0e40\u0e1b\u0e47\u0e19\u0e28\u0e32\u0e2a\u0e15\u0e23\u0e32\u0e08\u0e32\u0e23\u0e22\u0e4c\u0e17\u0e35\u0e48\u0e21\u0e2b\u0e32\u0e27\u0e34\u0e17\u0e22\u0e32\u0e25\u0e31\u0e22\u0e0b\u0e32\u0e23\u0e32\u0e15\u0e2d\u0e1f\u0e43\u0e19\u0e1b\u0e35 \u0e04.\u0e28. 1917 \u0e01\u0e48\u0e2d\u0e19\u0e08\u0e30\u0e01\u0e25\u0e31\u0e1a\u0e21\u0e32\u0e21\u0e2d\u0e2a\u0e42\u0e01\u0e43\u0e19\u0e1b\u0e35 \u0e04.\u0e28. 1922 \u0e41\u0e25\u0e30\u0e14\u0e33\u0e23\u0e07\u0e15\u0e33\u0e41\u0e2b\u0e19\u0e48\u0e07\u0e28\u0e32\u0e2a\u0e15\u0e23\u0e32\u0e08\u0e32\u0e23\u0e22\u0e4c\u0e14\u0e49\u0e32\u0e19\u0e17\u0e24\u0e29\u0e0e\u0e35\u0e1f\u0e31\u0e07\u0e01\u0e4c\u0e0a\u0e31\u0e19\u0e15\u0e31\u0e27\u0e41\u0e1b\u0e23\u0e40\u0e0a\u0e34\u0e07\u0e0b\u0e49\u0e2d\u0e19\u0e15\u0e25\u0e2d\u0e14\u0e0a\u0e35\u0e27\u0e34\u0e15\u0e17\u0e35\u0e48\u0e40\u0e2b\u0e25\u0e37\u0e2d \u0e1e\u0e24\u0e15\u0e34\u0e01\u0e23\u0e23\u0e21\u0e0a\u0e32\u0e22\u0e02\u0e2d\u0e1a\u0e02\u0e2d\u0e07\u0e1f\u0e31\u0e07\u0e01\u0e4c\u0e0a\u0e31\u0e19\u0e27\u0e34\u0e40\u0e04\u0e23\u0e32\u0e30\u0e2b\u0e4c \u0e07\u0e32\u0e19\u0e27\u0e34\u0e08\u0e31\u0e22\u0e0a\u0e34\u0e49\u0e19\u0e41\u0e23\u0e01\u0e02\u0e2d\u0e07\u0e1b\u0e23\u0e35\u0e27\u0e32\u0e25\u0e2d\u0e1f\u0e43\u0e19\u0e1b\u0e35 \u0e04.\u0e28. 1914 \u0e40\u0e01\u0e35\u0e48\u0e22\u0e27\u0e02\u0e49\u0e2d\u0e07\u0e01\u0e31\u0e1a\u0e2d\u0e19\u0e38\u0e01\u0e23\u0e21\u0e15\u0e31\u0e49\u0e07\u0e09\u0e32\u0e01\u0e41\u0e25\u0e30\u0e2a\u0e21\u0e01\u0e32\u0e23\u0e40\u0e0a\u0e34\u0e07\u0e1b\u0e23\u0e34\u0e1e\u0e31\u0e19\u0e18\u0e4c \u0e01\u0e48\u0e2d\u0e19\u0e08\u0e30\u0e02\u0e22\u0e31\u0e1a\u0e44\u0e1b\u0e28\u0e36\u0e01\u0e29\u0e32\u0e04\u0e38\u0e13\u0e2a\u0e21\u0e1a\u0e31\u0e15\u0e34\u0e02\u0e2d\u0e07\u0e2d\u0e19\u0e38\u0e01\u0e23\u0e21\u0e1f\u0e39\u0e23\u0e34\u0e40\u0e22\u0e23\u0e4c \u0e41\u0e15\u0e48\u0e04\u0e27\u0e32\u0e21\u0e2a\u0e19\u0e43\u0e08\u0e2b\u0e25\u0e31\u0e01\u0e17\u0e35\u0e48\u0e17\u0e33\u0e43\u0e2b\u0e49\u0e0a\u0e37\u0e48\u0e2d\u0e02\u0e2d\u0e07\u0e40\u0e02\u0e32\u0e40\u0e1b\u0e47\u0e19\u0e17\u0e35\u0e48\u0e08\u0e14\u0e08\u0e33\u0e04\u0e37\u0e2d 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1913 \u0e20\u0e32\u0e22\u0e43\u0e15\u0e49\u0e01\u0e32\u0e23\u0e14\u0e39\u0e41\u0e25\u0e02\u0e2d\u0e07\u0e14\u0e21\u0e34\u0e17\u0e23\u0e35 \u0e40\u0e22\u0e01\u0e2d\u0e23\u0e2d\u0e1f (Dmitri Egorov) \u0e41\u0e25\u0e30\u0e23\u0e48\u0e27\u0e21\u0e2a\u0e21\u0e31\u0e22\u0e01\u0e31\u0e1a\u0e19\u0e34\u0e42\u0e04\u0e44\u0e25 \u0e25\u0e39\u0e0b\u0e34\u0e19 (Nikolai Luzin) \u0e1c\u0e39\u0e49\u0e2d\u0e32\u0e22\u0e38\u0e21\u0e32\u0e01\u0e01\u0e27\u0e48\u0e32\u0e40\u0e02\u0e32\u0e40\u0e08\u0e47\u0e14\u0e1b\u0e35\u0e41\u0e15\u0e48\u0e01\u0e33\u0e25\u0e31\u0e07\u0e28\u0e36\u0e01\u0e29\u0e32\u0e2d\u0e22\u0e39\u0e48\u0e43\u0e19\u0e0a\u0e48\u0e27\u0e07\u0e40\u0e27\u0e25\u0e32\u0e40\u0e14\u0e35\u0e22\u0e27\u0e01\u0e31\u0e19 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\u0e04.\u0e28. 1914 \u0e40\u0e01\u0e35\u0e48\u0e22\u0e27\u0e02\u0e49\u0e2d\u0e07\u0e01\u0e31\u0e1a\u0e2d\u0e19\u0e38\u0e01\u0e23\u0e21\u0e15\u0e31\u0e49\u0e07\u0e09\u0e32\u0e01\u0e41\u0e25\u0e30\u0e2a\u0e21\u0e01\u0e32\u0e23\u0e40\u0e0a\u0e34\u0e07\u0e1b\u0e23\u0e34\u0e1e\u0e31\u0e19\u0e18\u0e4c \u0e01\u0e48\u0e2d\u0e19\u0e08\u0e30\u0e02\u0e22\u0e31\u0e1a\u0e44\u0e1b\u0e28\u0e36\u0e01\u0e29\u0e32\u0e04\u0e38\u0e13\u0e2a\u0e21\u0e1a\u0e31\u0e15\u0e34\u0e02\u0e2d\u0e07\u0e2d\u0e19\u0e38\u0e01\u0e23\u0e21\u0e1f\u0e39\u0e23\u0e34\u0e40\u0e22\u0e23\u0e4c \u0e41\u0e15\u0e48\u0e04\u0e27\u0e32\u0e21\u0e2a\u0e19\u0e43\u0e08\u0e2b\u0e25\u0e31\u0e01\u0e17\u0e35\u0e48\u0e17\u0e33\u0e43\u0e2b\u0e49\u0e0a\u0e37\u0e48\u0e2d\u0e02\u0e2d\u0e07\u0e40\u0e02\u0e32\u0e40\u0e1b\u0e47\u0e19\u0e17\u0e35\u0e48\u0e08\u0e14\u0e08\u0e33\u0e04\u0e37\u0e2d \u0e1e\u0e24\u0e15\u0e34\u0e01\u0e23\u0e23\u0e21\u0e0a\u0e32\u0e22\u0e02\u0e2d\u0e1a\u0e02\u0e2d\u0e07\u0e1f\u0e31\u0e07\u0e01\u0e4c\u0e0a\u0e31\u0e19\u0e27\u0e34\u0e40\u0e04\u0e23\u0e32\u0e30\u0e2b\u0e4c (boundary properties of...\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathed.bru.ac.th\/en\/2026\/07\/13\/\u0e2d\u0e35\u0e27\u0e32\u0e19-\u0e1b\u0e23\u0e35\u0e27\u0e32\u0e25\u0e2d\u0e1f-\u0e19\u0e31\u0e01\u0e04\u0e13\u0e34\u0e15\u0e28\/\" \/>\n<meta property=\"og:site_name\" content=\"\u0e2a\u0e32\u0e02\u0e32\u0e27\u0e34\u0e0a\u0e32\u0e04\u0e13\u0e34\u0e15\u0e28\u0e32\u0e2a\u0e15\u0e23\u0e4c\" \/>\n<meta property=\"article:published_time\" content=\"2026-07-13T04:16:54+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathed.bru.ac.th\/wp-content\/uploads\/2026\/07\/ChatGPT-Image-Jul-13-2026-11_15_55-AM.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1536\" \/>\n\t<meta property=\"og:image:height\" content=\"1024\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"CM Admin\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"CM Admin\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" 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