Extending the eleven functions to the complex numbers

This section lists the formulas used for evaluating the eleven functions when their arguments or their values are complex numbers. The idea is to replace the expression on the left side of the formula by the expression on the right. This new expression contains only functions of real numbers. (Assume that x = a + b i  is a complex number in rectangular form and that y = r ∠ s = r e i s  is another complex number in polar or exponential form.)
  1. To evaluate exp (x) or e x, where x is complex, use the following formula. The right side contains only evaluations of exp, sin or cos of real numbers:
  2. To evaluate sin (x), where x is complex, use the following formula. The two terms in the numerator on the right side are evaluated using formula # 1:
  3. To evaluate cos (x), where x is complex, use the following formula. The two terms in the numerator on the right side are evaluated using formula # 1:
  4. To evaluate tan (x), where x is complex, use the following formula. The numerator on the right side is evaluated using formula # 2 and the denominator using formula # 3:
  5. To evaluate the square root of a complex number y, first express y in polar form, and then use this formula:
  6. To evaluate the natural logarithm of a complex number y, first express y in polar form or exponential form, and then use this formula:
  7. To evaluate the base 10 logarithm of a complex number y use the following formula. The numerator on the right side is evaluated using formula # 6:
  8. To evaluate the power function, y x, where x and y are complex, first express y in polar form or exponential form and then use the following formula. Click here to see an example:
  9. To evaluate the arcsin of a complex number a + b i use this formula:
    where
    and
  10. To evaluate the arccos of a complex number a + b i use this formula:
    where A and B are the same as for formula # 9.

  11. To evaluate the arctan of a complex number a + b i use this formula:




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