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Exponential functions occur very often in solutions of differential equations. The exponential functions can be defined as solutions of differential equations. Indeed, the exponential function is a solution of the simplest possible differential equation, namely ′ = .
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Here φ is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counterclockwise and in radians. The original proof is based on the Taylor series expansions of the exponential function e z (where z is a complex number) and of sin x and cos x for real numbers x ( see above ).
Complex exponential function: The exponential function exactly maps all lines not parallel with the real or imaginary axis in the complex plane, to all logarithmic spirals in the complex plane with centre at : () = (+) + ⏟ = + = ( + ) ⏟ The pitch angle of the logarithmic spiral is the angle between the line and the imaginary axis.
Figure 1 illustrates how this looks. It presents two plots generated using 10,000 simulated points. The left plot, titled 'Concave Line with Log-Normal Noise', displays a scatter plot of the observed data (y) against the independent variable (x). The red line represents the 'Median line', while the blue line is the 'Mean line'.
The solutions to the differential equation are a family of functions. Graphically, this can be plotted in the phase plane like a two-dimensional vector field. Vectors representing the derivatives of the points with respect to a parameter (say time t), that is (dx/dt, dy/dt), at representative points are drawn.
Parameter plane of the complex exponential family f(z)=exp(z)+c with 8 external ( parameter) rays. In the theory of dynamical systems, the exponential map can be used as the evolution function of the discrete nonlinear dynamical system. [1]
Usually, this estimator is the proportion of times that the number occurs in the data set. If the points in the plot tend to converge to a straight line for large numbers in the x axis, then the researcher concludes that the distribution has a power-law tail. Examples of the application of these types of plot have been published. [61]