Functions of Complex Numbers

Once we have mastered using i as an exponent, we can advance to functions of complex numbers.  Exponential functions, of course, are simply functions which use i as an exponent.

Logarithm functions

Since a complex number can be written in exponential form as r times e to the power i times P, a logarithm is almost immediate.  We get natural logarithm of r times e to the power i P equals the natural logarithm of r, plus i times P, which gives the principal value of the natural logarithm.  But the angle P is not unique, as it can vary by multiples of 2 pi.  Therefore, the formula for the natural logarithm of a complex number is natural logarithm of r, plus i times P, plus 2 k pi i.

For example, we can find the natural logarithm of i.  Its principal value is one half pi times i .

Another interesting example is the natural logarithm of negative one.  Its principal value is pi times i.  (And you thought you couldn't take logarithms of negative numbers!  You can, but the answers are not real numbers.)

We can use the change-of-base formula to find other base logarithms (including complex bases).  For example, logarithm of e, base i, equals negative 2 i divided by pi.

Trigonometric Functions

Because of the formula e to the power i x equals cosine x plus i times sine x, we can immediately obtain some formulas for the sine and cosine functions, in terms of complex exponentials.  These formulas are often used as definitions of the trigonometric functions for complex numbers.

cosine x equals one half the sum of e to the power i x and e to the power negative i x, sine x equals e to the power i x minus e to the power negative i x, all divided by 2 i

Substituting the quantity ix in place of the variable x, we can produce formulas for pure imaginary numbers.  Notice that the results are related to the hyperbolic functions.

cosine of i x equals the hyperbolic cosine of x, sine of i x equals i times the hyperbolic sine of x

Then, using the formulas for the sine and cosine of the sum of two angles, we can obtain formulas for the sine and cosine of complex numbers.

cosine a plus b i equals cosine a hyperbolic cosine b plus i sine a hyperbolic sine b
sine a plus b i equals sine a hyperbolic cosine b plus i cosine a hyperbolic sine b

As an example, consider the cosine of i.  We get cosine of i equals the hyperbolic cosine of one

Hyperbolic Functions

Using the connection between hyperbolic functions and trigonometric functions, the results for hyperbolic functions are almost immediate.

hyperbolic cosine a plus b i equals hyperbolic cosine a cosine b plus i hyperbolic sine a sine b
hyperbolic sine a plus b i equals hyperbolic sine a cosine b plus i hyperbolic cosine a sine b

As an example, consider the hyperbolic cosine of i.  We get hyperbolic cosine of i equals the cosine of one.

Inverse Trigonometric Functions

Since cosine x equals e to the power i x, plus e to the power negative i x, all divided by 2, we can find the inverse by replacing the function with the variable y, swapping x and y, then solving for y.  After the variable swap, we have x equals e to the power i y, plus e to the power negative i y, all divided by 2.  Multiplying both sides by 2 times e to the power i y gives the equation zero equals e to the power 2 i y, minus 2 x e to the power i y, plus 1.  This can be solved using the quadratic formula, to get e to the power i y equals the fraction 2 x plus or minus the square root of 4 x squared minus 4, all over 2.  After simplifying, we can take the natural logarithm of both sides and divide by i.  Then, the result is inverse cosine of x equals negative i times the natural logarithm of x plus or minus i times the square root of 1 minus x squared.  (A factor of i typically appears as the coefficient of the square root so that the similarities between the inverse cosine and inverse sine formulas are more apparent.)

By similar arguments, the inverse sine formula is found to be inverse sine x equals negative i times the natural logarithm of i x plus or minus the square root of 1 minus x squared, and the inverse tangent formula is one half i times the natural logarithm of i plus x over i minus x.

As an example, consider the inverse cosine of i. We shall find only one of its many values, by using the plus sign in the formula.  We get:negative i times the natural logarithm of i times 1 plus the square root of 2  Then we use the logarithm formula to continue:negative i times the quantity natural logarithm of 1 plus square root of 2, plus pi over 2 times i This can be simplified, and we get Pi over 2 minus i times the natural logarithm of 1 plus the square root of 2.

Inverse Hyperbolic Functions

The formulas are identical to those previously found for inverse hyperbolic functions:

inverse hyperbolic cosine of x equals the natural logarithm of x plus or minus the square root of x squared minus one
inverse hyperbolic sine of x equals the natural logarithm of x plus or minus the square root of x squared plus one

For example, one of the values of the inverse hyperbolic cosine of i (using the plus sign in the formula) is:the inverse hyperbolic cosine of i equals the natural logarithm of i times the quantity one plus the square root of two.  Then using the logarithm formula, we get:the inverse hyperbolic cosine of i equals the natural logarithm of one plus square root of 2, plus pi over 2 times i.