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[parent] complex exponential function (Definition)

The complex exponential function $ \exp:\,\mathbb{C}\to \mathbb{C}$ may be defined in many equivalent ways: Let $ z = x+iy$ where $ x,\,y\in\mathbb{R}$.

  • $ \displaystyle\exp{z} := e^x(\cos{y}+i\sin{y})$
  • $ \displaystyle\exp{z} := \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n$
  • $ \displaystyle\exp{z} := \sum_{n = 0}^\infty\frac{z^n}{n!}$
The complex exponential function is usually denoted in power form:
$\displaystyle e^z := \exp{z},$
where $ e$ is the Euler number. It also coincincides with the real exponential function when $ z$ is real (choose $ y = 0$). It has all the properties of power, e.g. $ e^{-z} = \frac{1}{e^z}$; these are consequences of the addition formula
$\displaystyle e^{z_1+z_2} = e^{z_1}e^{z_2}$
of the complex exponential function.

The function gets all complex values except 0 and is periodic having the prime period (the period with least non-zero modulus) $ 2\pi i$. The $ \exp$ is holomorphic, its derivative

$\displaystyle \frac{d}{dz}e^z = e^z,$
which is obtained from the series form via termwise differentiation, is similar as in $ \mathbb{R}$.

So we have a fourth way to define

  • $ \exp{z} := w(z)$
with $ w$ the solution of the differential equation $ \displaystyle\frac{dw}{dz} = w$ under the initial condition $ w(0) = 1$.

Some formulae:

$\displaystyle \vert e^z\vert = e^x, \quad \arg{e^z} = y+2n\pi\quad(n = 0,\,\pm1,\,\pm2,\,\ldots),$
Re$\displaystyle (e^z) = e^x\cos{y},$   Im$\displaystyle (e^z) = e^x\sin{y}$



"complex exponential function" is owned by pahio.
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See Also: exponential function defined as limit of powers, exponential function, complex sine and cosine, proof of equivalence of formulas for exp, derivative of exponential function, convergence of Riemann zeta series

Also defines:  exponential function, prime period

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Attachments:
complex logarithm (Definition) by pahio
periodicity of exponential function (Theorem) by pahio
proof of addition formula of exp (Proof) by pahio
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Cross-references: initial condition, differential equation, solution, similar, termwise differentiation, series, derivative, holomorphic, modulus, complex, function, addition formula, consequences, properties, real, Euler number, power, equivalent
There are 25 references to this entry.

This is version 16 of complex exponential function, born on 2004-10-10, modified 2007-05-02.
Object id is 6341, canonical name is ComplexExponentialFunction.
Accessed 24042 times total.

Classification:
AMS MSC: 30D20 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Entire functions, general theory)
 32A05 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Power series, series of functions)
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