exponential-form

Exponential form

You now know how to do lots of operations with complex numbers: add, subtract, multiply, divide, raise to a power and even square root. The last operation to learn is raising the number e to the power of an imaginary number.

How do we calculate e to the power of an imaginary number?

• Given an imaginary number (iθ) we can define exponentiation as

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  • is the complex number with modulus 1 and argument θ

• This works with our current rules of exponents

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    • This shows e to the power 0 would still give the answer of 1

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    • This is because when you multiply complex numbers you can add the arguments

    • This shows that when you multiply two powers you can still add the indices

  • <img alt=”straight e to the power of iθ subscript 1 end exponent over straight e to the power of iθ subscript 2 end exponent equals straight e to the power of straight i left parenthesis straight theta subscript 1 minus straight theta subscript 2 right parenthesis end exponent” data-mathml='<math ><semantics><mrow><mfrac><msup><mi mathvariant=”normal”>e</mi><msub><mi>iθ</mi><mn>1</mn></msub></msup><msup><mi mathvariant=”normal”>e</mi><msub><mi>iθ</mi><mn>2</mn></msub></msup></mfrac><mo>=</mo><msup><mi mathvariant=”normal”>e</mi><mrow><mi mathvariant=”normal”>i</mi><mo>(</mo><msub><mi mathvariant=”normal”>θ</mi><mn>1</mn></msub><mo>-</mo><msub><mi mathvariant=”normal”>θ</mi><mn>2</mn></msub><mo>)</mo></mrow></msup></mrow><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“language”:”en”,”fontFamily”:”Times New Roman”,”fontSize”:”18″}</annotation></semantics></math>’ height=”61″ role=”math” src=”data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2261%22%20wid