Plot[Im@Exp[-I x], {x, -5, 5}, PlotStyle -> Thick,
ColorFunction -> "StarryNightColors"]Mathematica has a built-in function called Exp that can be used to calculate the exponential function. By inputting Exp[-I*θ], Mathematica will recognize it as e^-iθ and simplify it to cos(θ) - i*sin(θ). This is the Euler's identity.
Yes, you can also use the notation Power[E, -I*θ] to represent e^-iθ in Mathematica. This will also be recognized as the Euler's identity and simplified accordingly.
You can use the Euler's identity in your calculations by replacing e^-iθ with Exp[-I*θ] or Power[E, -I*θ]. This will allow you to simplify complex expressions and perform various mathematical operations.
Yes, Mathematica has a wide range of built-in functions and identities related to the Euler's identity. These include the TrigReduce function which simplifies trigonometric expressions, and the TrigExpand function which expands trigonometric expressions into their component terms.
You can plot the Euler's identity in Mathematica by using the Plot function and specifying the expression Exp[-I*θ] or Power[E, -I*θ] as the function to be plotted. This will graph the real and imaginary parts of the Euler's identity as a function of θ, creating a circle on the complex plane.