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Mathematics LibreTexts

1.7: The Exponential Function

  • Page ID
    47203
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    We have Euler's formula: \(e^{i\theta} = \cos (\theta) + i \sin (\theta)\). We can extend this to the complex exponential function \(e^z\).

    Definition

    For \(z = x + iy\) the complex exponential function is defined as

    \(e^z = e^{x + iy} = e^x e^{iy} = e^x (\cos (y) + i \sin (y))\).

    In this definition \(e^x\) is the usual exponential function for a real variable \(x\).

    It is easy to see that all the usual rules of exponents hold:

    1. \(e^0 = 1\)
    2. \(e^{z_1 + z_2} = e^{z_1} e^{z_2}\)
    3. \((e^z)^n = e^{nz}\) for positive integers \(n\).
    4. \((e^z)^{-1} = e^{-z}\)
    5. \(e^z \ne 0\)
      It will turn out that the property \(\dfrac{de^z}{dz} = e^z\) also holds, but we can’t prove this yet because we haven’t defined what we mean by the complex derivative \(\dfrac{d}{dz}\).
      Here are some more simple, but extremely important properties of \(e^z\). You should become fluent in their use and know how to prove them.
    6. \(|e^{i \theta}| = 1\)
      Proof.
      \(|e^{i \theta}| = |\cos (\theta) + i \sin (\theta)| = \sqrt{\cos ^2 (\theta) + \sin ^2 (\theta)} = 1\)
    7. \(|e^{x + iy}| = e^x\) (as usual \(z = x + iy\) and \(x, y\) are real).
      Proof. You should be able to supply this. If not: ask a teacher or TA.
    8. The path \(e^{it}\) for \(0 < t < \infty\) wraps counterclockwise around the unit circle. It does so infinitely many times. This is illustrated in the following picture.

    屏幕快照 2020-08-14 上午2.49.38.png
    The map \(t \to e^{it}\) wraps the real axis around the unit circle.

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