Complex Exponential Function

A Complex Exponential Function is an exponential function based on the complex exponentiation operation of the form [math]\displaystyle{ f(x)=e^{ix} }[/math].

• AKA: Euler's Formula, CEF.
• Context:
  • For any real number, [math]\displaystyle{ x }[/math], it can be defined by the following power series: [math]\displaystyle{ e^{ix} = 1 + x + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \dots = \underbrace{1 - \frac{(x)^2}{2!} +\frac{x^4}{4!} +\dots}_\textrm{cosine function}+i\underbrace{( x-\frac{x^3}{3!} + \dots)}_\textrm{sine function} }[/math] Thus,: [math]\displaystyle{ e^{ix}=cosx+isinx }[/math] this mathematical relationship is called Euler's Formula,
  • Any complex number such as [math]\displaystyle{ z= a+ib }[/math] can be written into its complex exponential form: [math]\displaystyle{ z=re^{i\theta} }[/math] where [math]\displaystyle{ r=|z| }[/math] is the complex number modulus and [math]\displaystyle{ \theta=arg(z) }[/math] is the argument.
• Example(s):
  • Second-Order Differential Equation Solution.
  • Fourier Transform.
  • Spherical Harmonics.
  • …
• Counter-Example(s):
  • Real Numbers.
  • Real Exponential Function.
• See: Exponential Function, Natural Exponential Function, Complex Number, Complex Roots, Complex Analysis.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Complex_number Retrieved:2015-21-11
  • QUOTE: Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponential function [math]\displaystyle{ exp(z) }[/math], also written [math]\displaystyle{ e^z }[/math], is defined as the infinite series

[math]\displaystyle{ \exp(z):= 1+z+\frac{z^2}{2\cdot 1}+\frac{z^3}{3\cdot 2\cdot 1}+\cdots = \sum_{n=0}^{\infty} \frac{z^n}{n!}. \, }[/math]

and the series defining the real trigonometric functions sine and cosine, as well as hyperbolic functions such as sinh also carry over to complex arguments without change. Euler's identity states:

[math]\displaystyle{ \exp(i\varphi) = \cos(\varphi) + i\sin(\varphi) \, }[/math]

for any real number [math]\displaystyle{ \varphi }[/math], in particular [math]\displaystyle{ \exp(i \pi) = -1 \, }[/math].

Unlike in the situation of real numbers, there is an infinitude of complex solutions [math]\displaystyle{ z }[/math] of the equation [math]\displaystyle{ \exp(z) = w \, }[/math] for any complex number [math]\displaystyle{ w \ne 0 }[/math].

2014

• http://mathfaculty.fullerton.edu/mathews/c2003/ComplexFunExponentialMod.html

2013

• (Serge Lang, 2013) ⇒ Lang, S.(2013). Complex analysis. Vol. 103. Springer Science & Business Media ⇒ http://books.google.com/books?id=deQlBQAAQBAJ
  • QUOTE: We may therefore compare our series with a geometric series to get the absolute uniform convergence.

    The series

[math]\displaystyle{ e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!} }[/math]

Therefore defines a continuous function for all values of [math]\displaystyle{ z }[/math]. Similarly, the series

[math]\displaystyle{ sin z= z-\frac{(z)^3}{3!} + \frac{(z)^5}{5!} \dots=\sum_{n=0}^{\infty} (-1)^{n}\frac{z^{2n+1}}{(2n+1)!} }[/math]

and

[math]\displaystyle{ cos z= 1 - \frac{(z)^2}{2!} +\frac{z^4}{4!} +\dots=\sum_{n=0}^{\infty} (-1)^{n}\frac{z^{2n}}{(2n)!} }[/math]

converge absolutely and uniformly for all [math]\displaystyle{ |z|\leq r }[/math]. They give extensions of the sine and cosine functions to complex numbers. We shall see later that [math]\displaystyle{ exp(z)=e^z }[/math] as defined in Chapter I, and these series the unique analytic functions which coincide with the usual exponential, sine, and cosine functions, respectively, when [math]\displaystyle{ z }[/math] is real.

1999

• (Wolfram Math, 1999) ⇒ http://mathworld.wolfram.com/ExponentialFunction.html Retrieved:2015-11-21.
  • QUOTE: The exponential function is the entire function defined by

[math]\displaystyle{ exp(z)=e^z }[/math]

where [math]\displaystyle{ e }[/math]is the solution of the equation [math]\displaystyle{ \int_1^xdt/t }[/math] so that [math]\displaystyle{ e=x=2.718\dots exp(z) }[/math] is also the unique solution of the equation [math]\displaystyle{ df/dz=f(z) }[/math] with [math]\displaystyle{ f(0)=1 }[/math].

The exponential function is implemented in the Wolfram Language as Exp[z].

It satisfies the identity

[math]\displaystyle{ exp(x+y)=exp(x)exp(y) }[/math]

If [math]\displaystyle{ z=x+iy }[/math]

[math]\displaystyle{ e^z=e^{x+iy}=e^xe^{iy}=e^x(cosy+isiny) }[/math]

1976

• Fischer, Gerd. Complex analytic geometry. Vol. 538. Springer, 1976.

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