Graphing a wave in complex coordinate2/17/2024 ![]() ![]() This graphical representation of the complex exponential function, clearly shows the relation to the trigonometric function the real number part of is cosine and the imaginary part of is sine function with the period of in radians. ![]() Graph of Real number part of Imaginary number part of And, when it is projected to the real number (top view) and imaginary number axis (side view), it becomes a trigonometric function, respectively cosine and sine. Surprisingly, it is a spiral spring (coil) shape, rotating around a unit circle. The following images show the graph of the complex exponential function,, by plotting the Taylor series of in the 3D complex space (x - real - imaginary axis). We know that the exponential function, is increasing exponentially as x grows. Now, we find out equals to, which is known as Euler's Equation. Finally, Leonhard Euler completed this relation by bringing the imaginary number, into the above Taylor series instead of and instead of. Mathematicians had tried to figure out this weird relationship between the exponential function and the sum of 2 oscillating functions. But, what does this exponential function have to do with periodic (oscillating) functions, and ? As you know, the exponential function, increases exponentially as input x grows. Notice is almost identical to Taylor series of all terms in the series are exactly same except signs. Euler's equation (formula) shows a deep relationship between the trigonometric function and complex exponential function.įirst, take a look the Taylor series representation of exponential function, and trigonometric functions, sine, and cosine. Euler's equation is one of most remarkable and mysterious discoveries in Mathematics. ![]()
0 Comments
Leave a Reply.AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |