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In control theory, overshoot refers to an output exceeding its final, steady-state value. [2] For a step input, the percentage overshoot (PO) is the maximum value minus the step value divided by the step value. In the case of the unit step, the overshoot is just the maximum value of the step
For a step input, the percentage overshoot (PO) is the maximum value minus the step value divided by the step value. In the case of the unit step, the overshoot is just the maximum value of the step response minus one. The percentage overshoot (PO) is related to damping ratio (ζ) by:
Tay, Mareels and Moore (1998) defined settling time as "the time required for the response curve to reach and stay within a range of certain percentage (usually 5% or 2%) of the final value." [ 2 ] Mathematical detail
The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane.
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function = =. Zeta functions include: Airy zeta function, related to the zeros of the Airy function; Arakawa–Kaneko zeta function; Arithmetic zeta function
The local zeta function Z(X, t) is viewed as a function of the complex variable s via the change of variables q −s. In the case where X is the variety V discussed above, the closed points are the equivalence classes x=[P] of points P on V ¯ {\displaystyle {\overline {V}}} , where two points are equivalent if they are conjugates over F .
Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y 1/2 /log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x.
Like the Riemann zeta function, the multiple zeta functions can be analytically continued to be meromorphic functions (see, for example, Zhao (1999)). When s 1, ..., s k are all positive integers (with s 1 > 1) these sums are often called multiple zeta values (MZVs) or Euler sums. These values can also be regarded as special values of the ...