Complex Exponential Function

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



References

2015

[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

2013

[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

[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.