Solving Complex Equation Examples Example 3: Solve for h. {eq}5i = (6hi + 2)(5i - 8) {/eq} 1) This is a complex math equation that involves complex numbers and an imaginary number. The complex numbers are {eq}(6hi + 2) {/eq} and {eq}(5i - 8) {/eq}. The purely imagina...
The equation of a hyperbola with a horizontal axis is (x2/ a2) - (y2 / b2) = 1 where a and b are positive constants. A circle has a constant distance from the center point, while a hyperbola is a curve that has two focus points (+ae, 0), and (-ae, 0). What is the diff...
Note that there is a minus sign in the real part since, at some point, we faced a multiplication of two imaginary numbers i ⋅ ii⋅i, which equals −1−1 by definition. Multiplying complex numbers isn't that scary. Is it? So what about dividing complex numbers? Let...
In other words, (6) can be thought of as the output equation of a dynamic system. Unfortunately, the term 𝑒−𝑗2𝜃𝑠 provides a time-varying characteristic to the dynamic model. Another difficulty to overcome is that the sequence components, i.e., the state of the system, ...
The equation equals Eq. (1.189), but the form is not divided into two conjugated eigenvalues, which are now enclosed in the expanded range 2N, so that (3.39)λr=λr+N⁎ and (3.40)Rr=Rr+N⁎. The impulse response based on the model (3.38) is (see also Eq. (1.91)) (3.41)h(...
When the roots are complex numbers—λ1=p+qi and λ2=p−qi, where p and q are real numbers—the two corresponding solutions of the differential equation ay″+by′+cy=0 are y1(t)=e(p+qi)t and y2(t)=e(p−qi)t. At this point, a crucial fact to know is Euler's formula,...
In the context of complex numbers, a logarithm of the complex number z is any complex number w such that ew = z. This equation has no solution if z = 0, and it has infinitely many solutions otherwise: for any solution w, w + 2nπi is also a solution for all integers n. Solution...
Definition:The imaginary unit i is the solution to the equation x2+1=0. Imaginary unit has the property i2=-1 Let’s try to see what numbers we can get by raising the imaginary unit to other powers: i0=1 (Every number raised to zero power equals to 1) ...
Sign up with one click: Facebook Twitter Google Share on Facebook complex notation [′käm‚pleks nō′tā·shən] (physics) The representation of a physical quantity by a complex number whose real component equals the instantaneous value of the physical quantity, a sinusoidally varying quan...
It is clear that the equation cosh2θ−sinh2θ=1 holds, which reminds us of the equation of a hyperbola, x2−y2=1. This is why they are called hyperbolic functions. Note that cosθ is an even (symmetric) function of θ, so changing the sign of θ should not change...