Euler’s identity

We now repeat the problem with the s = jco substitution in the characteristic equation, and rewrite the time delay function with Euler s identity ... 

Apply Euler s identity and the final result for the normalized response is V (t)l... 

Economics in process control, 3, 10-11, 15, 26, 532-34 Environmental regulations, 3 Equal-percentage valve, 254, 255 Equations of state, 57 Equilibria, 56, 78 chemical, 56 phase, 56-57, 71, 75, 78 Error criteria (see Time integral criteria) Euler s identities, 131-32, 149 Experimental modeling, 45, 656 frequency response techniques, 668 process identification, 657-62 time constant determination, 228, 232 Exponential function, 130 approximations, 215-16 Laplace transform, 130 z-transform, 592... 

In the relationship above we have used Euler s identity (see Section 7.2) to express sin ((onT) ... 

To prove eq. (28.9), proceed along the same lines as above, using the following Euler s identity ... 

There is a near-mystical expression of equality in mathematics known as Euler s Identity, which links trigonometry, exponential functions, and complex numbers in a single equation ... 

Converting the cosine/sine form to the complex exponential form allows many manipulations that would be very difficult otherwise (for an example, see Section 2 of Appendix A Convolution and DFT Properties). But, if you re totally uncomfortable with complex numbers and Euler s Identity (or with the identities of IS"" century mathematicians in general), then you can write the DFT in real number terms as a form of the Fourier Series ... 

The well-known Euler s identity is a convenient conversion of the polar and Cartesian forms into an exponential form, given by... 

The reciprocal is also true if a function f(x,y,z) obeys Euler s identity, it is homogeneous with a degree of m with regard to variables x, y and z. 

Euler s Identities These identities allow sines and cosines to be expressed in terms of complex numbers, and vice versa ... 

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