1. Where " g " is an arbitrary function for growth rate. 2. These circuits can be used to compute arbitrary functions of the initial configuration. 3. Where the are completely arbitrary functions of time. 4. The Sum function can be employed for arbitrary functions merely by employing the appropriate expressions. 5. I can factor any arbitrary function of x and y out of any of the terms. 6. The Hankel transform can express a fairly arbitrary function as an integral of Bessel functions of different scales. 7. Where \ lambda is an arbitrary complex constant, and U ( \ lambda ) is an arbitrary function . 8. Note that Z [ J ] as originally defined, is completely independent of the arbitrary function \ chi ( x ). 9. This is a very practical challenge for algorithms that are meant to find zeros of functions, whether arbitrary functions or simply polynomials. 10. The skills required to solve a quadratic equation on a calculator are in fact applicable to finding the real roots of any arbitrary function .
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