- #1
Likemath2014
- 17
- 0
How can we show that the following equation has infinitely many solutions
[tex]e^z-z^2=0[/tex].
Thanks
[tex]e^z-z^2=0[/tex].
Thanks
A complex exponential equation is an equation that involves a complex number raised to a power. It can be written in the form of z = reiθ, where z is a complex number, r is the magnitude, and θ is the argument or angle.
Some properties of complex exponential equations include the ability to use the laws of exponents, such as the product and quotient rules, to simplify expressions. Additionally, complex exponential equations can be graphed in the complex plane, and they have periodicity and symmetry properties.
To solve a complex exponential equation, first write the equation in the form of z = reiθ. Then, use the laws of exponents to manipulate the equation and isolate the complex number z. Finally, use the inverse of the exponential function, ez = excos(y) + iexsin(y), to find the values of x and y.
Complex exponential equations have many applications in physics, engineering, and other fields. For example, they can be used to model the behavior of electrical circuits, analyze vibrations in mechanical systems, and describe quantum mechanical systems.
Complex exponential equations and trigonometric functions are closely related. In fact, the real and imaginary parts of a complex exponential equation can be expressed in terms of sine and cosine functions. This relationship is known as Euler's formula: eix = cos(x) + isin(x).