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A thorough discussion and development of the calculus of real-valued functions of complex-valued vectors is given using the framework of the Wirtinger Calculus.
“Zur Formalen Theorie der Funktionen von mehr Complexen Veränderlichen,” W. Wirtinger, Math. Ann., 97: 357-75, 1927
1927
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Complex Variables and the Laplace Transform for Engineers, W. LePage, 1980/1961, Dover Publications, New York
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Introduction to Complex Analysis, Z. Nehari, 1961, Allyn & Bacon, Inc
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An Introduction to Complex Analysis in Several Variables, Revised 3rd Edition, Lars Hörmander, 1990, North-Holland. (First Edition was published in 1966.)
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Optimization by Vector Space Methods, D.G. Luenberger, Wiley, 1969
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Introduction to Adaptive Arrays, R. Monzingo & T. Miller, 1980, Wiley, New York
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Geometrical Methods of Mathematical Physics, Bernard Schutz,1980, Cambridge University Press
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Linear Operator Theory in Engineering and Science, A.W. Naylor & G.R. Sell, Springer-Verlag, 1982
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Complex Variables: Harmonic and Analytic Functions, F. Flanigan, 1972/1983, Dover Publications, New York
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“A Complex Gradient Operator and its Application in Adaptive Array Theory,” D.H. Brandwood, IEE Proceedings H (Microwaves, Optics, and Antennas) (British), Vol. 130, No. 1, pp. 11-16, Feb. 1983
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Theory of Functions on Complex Manifolds, G.M. Henkin & J. Leiterer, 1984, Birkäuser
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Digital Communication, E. Lee & D. Messerchmitt, 1988, Kluwer, Boston
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Theory of Complex Functions, R. Remmert, 1991, Springer-Verlag
1991
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Adaptive Filter Theory, 3rd Edition, S. Haykin, 1991, Prentice-Hall, New Jersey. Especially Appendix B: “Differentiation with Respect to a Vector.”
1991
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1991
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Introduction to Complex Analysis, Part II: Functions of Several Variables, B.V. Shabat, American Mathematical Society, 1992
1992
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Fundamentals of Statistical Signal Processing: Estimation Theory , S.M. Kay, 1993, Prentice-Hall, New Jersey. Chapter 15: “Extensions for Complex Data and Parameters.”
1993
“Complex Digital Networks: A Sensitivity Analysis Based on the Wirtinger Calculus,” D. Franken, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 44(9):839-43, November 1997
1997
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“Natural Gradient Works Efficiently in Learning,” Shun-ichi Amari, Neural Computation, 10:251-76, 1998
1998
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“The Real-Complex Normal Distribution,” A. van den Bos, IEEE Transactions on Information Theory , 44(4):537-39, July 1998
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Complex Variables, 2nd Edition, S. Fisher, 1990/1999, Dover Publications, New York
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“A Newton-Like Algorithm for Complex Variables with Applications in Blind Equalization,” G. Yan & H. Fan, IEEE Proc. Signal Processing, 48(2):553-6, February 2000
2000
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Cited alongside, same era.
“Complex Gradient and Hessian,” A. van den Bos, IEE Proc.-Vis. Image Signal Processing (British), 141(6):380-82, December 1994
1994
Cited alongside, same era.
“A Cramér-Rao Lower Bound for Complex Parameters,” A. van den Bos, IEEE Transactions on Signal Processing , 42(10):2859, October 1994
1994
Cited alongside, same era.
Elements of Signal Detection & Estimation, C.W. Helstrom, 1995, Prentice Hall
1995
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“Estimation of Complex Fourier Coefficients,” A. van den Bos, IEE Proc.-Control Theory Appl. (British), 142(3):253-6, May 1995
1995
Cited alongside, same era.
“The Multivariate Complex Normal Distribution-A Generalization,” A. van den Bos, IEEE Transactions on Information Theory , 41(2):537-39, March 1995
1995
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Complex Variables and Applications, 6nd Edition, J. Brown & R. Churchill, 1996 McGraw-Hill, New York
1996
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“Price’s Theorem for Complex Variates,” A. van den Bos, IEEE Transactions on Information Theory , 42(1):286-7, January 1996
1996
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Principles of Mobile Communication, 2nd Edition, G.L. Stuber, 2001, Kluwer, Boston
2001
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Several Complex Variables, Corrected 2nd Edition, 2001, S.G. Krantz, AMS Chelsea Publishing
2001
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The Geometry of Physics, An Introduction, (with corrections and additions), Theodore Frankel, 2001, Cambridge University Press
2001
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Optimum Array Processing, H.L. Van Trees, 2002, Wiley Interscience
2002
Later among the works it cites.
Precoding and Signal Shaping for Digital Transmission , Robert Fischer, 2002, Wiley-Interscience. Especially Appendix A: “Wirtinger Calculus.”
2002
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Fundamentals of Adaptive Filtering, A.H. Sayed, 2003, Wiley
2003
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“Cramér-Rao Lower Bound for Constrained Complex Parameters,” A.K. Jagannatham & B.D. Rao, IEEE Signal Proc. Letters, 11(11):875-8, November 2004
2004
Later among the works it cites.