# Differential calculus and holomorphy by Jean Francois Colombeau

By Jean Francois Colombeau

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The quantity //*>,(/) is the frequency response function of a constant-parameter linear system between x(t) and y(t). 15, where an input x(f) produces many outputs y,<0. / = 1, 2, 3,.... Any output y,(0 is the result of x(t) passing through a constant-parameter linear system described by the frequency response function H (f). The noise terms n,{t) represent unmeasured extraneous output noise at the different outputs. 16 can be considered as a combination of separate single-input/single-output models.

B) damping ratio ζ. 6 \H{f)\. 2. 25). 8 2 η The half-power point bandwidth of a resonant physical system is defined as B =fi-fu where | H ( / i ) | = \H(f )\ = \\H{f )\ . 27). 1 is displaced from its position of equilibrium and then released. 10 Γ ? 0. 4. Determine the values of the analogous circuit parameters in terms of the mechanical parameters k, c, and m. REFERENCES 1. Bendat, J. , Nonlinear System Techniques and Applications, Wiley-Interscience, New York, 1998. 2. Stokey, W. , "Vibrations of Systems Having Distributed Mass and Elasticity," Chapter 7 in Shockand Vibration Handbook, 4th ed.

For example, if an experiment is repeated many times to yield results φ , i = 1, 2, . . 13) »r (=1 This expected value may or may not equal the true value φ. If it does, the estimate φ is said to be unbiased. Otherwise, it is said to be biased. 14) 22 BASIC DESCRIPTIONS AND PROPERTIES It follows that the bias error is a systematic error that always occurs with the same magnitude in the same direction when measurements are repeated under identical circumstances. The variance of the estimate, denoted by Var [], is defined as the expected value of the squared differences from the mean value.