Tables of Phase of a Semi-Infinite Unit 

 Attenuation Slope 



By D. E. THOMAS 



(Manuscript received February 24, 1956) 



Five and seven place tables of the integral 



B(x,) = ' log 



1 +a: 

 1 — X 



dx 



X 



which gives the 'phase associated with a semi-infinite unit slope of attenua- 

 tion, are now available in monograph form. The usefulness of this integral 

 and its tabulation are discussed. 



H. W. Bode' has shown that on the imaginary axis, the vahies of the 

 imaginary part of certain functions of a complex variable may be ob- 

 tained from the corresponding values of the real part, and vice versa. 

 This theorem was immediately recognized as a powerful tool in the com- 

 munications and network fields. The most generally useful function which 

 was given by Bode for use in applying this theorem to the solution of 

 communications problems, is the phase associated with a semi-infinite 

 unit slope of attenuation. This is given by the integral 



1 r':=Xc 

 5(.T.) = - log 



1 -\-x 



(J/Jy / 1 \ 



X 



1 - X 



where: 5(:i-c) is the phase in radians at frequency /c , 



x = ^ ,x, = ^^ < 1.0 



Jo Jo 



and fo = the frequency at which the semi-infinite unit slope 

 begins 



The usefulness of Integral (1) is illustrated by some of the communica- 

 tion problems which stimulated its accurate tabulation. 



iT 



1 Bode, H. W., Network Aiuily.sis and Feedback Amplifier Design, D. Van Nos- 

 trand Co., Inc., New York, 1945, Chap. XIV. 



2 Ibid: Chap. XV, pp. 342-343. 



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