How to calculate the exponential of a complex number?

To calculate the exponential of a complex number, use Euler's formula and the properties of exponents.

Euler's formula states that e^(ix) = cos(x) + i sin(x), where i is the imaginary unit. To find e^(a+bi), we can write it as e^a * e^(bi). Using Euler's formula, e^(bi) = cos(b) + i sin(b). Therefore, e^(a+bi) = e^a * (cos(b) + i sin(b)).

For example, to find e^(2+3i), we have e^2 * (cos(3) + i sin(3)). Using a calculator, we can find that cos(3) = -0.99 and sin(3) = 0.14. Therefore, e^(2+3i) = e^2 * (-0.99 + 0.14i).

To simplify expressions involving complex exponentials, we can use the properties of exponents. For example, e^(a+bi) * e^(c+di) = e^((a+c)+(b+d)i). This means that multiplying two complex exponentials adds their real and imaginary parts separately.

For example, to find e^(2+3i) * e^(4-2i), we can first find e^(2+3i) and e^(4-2i) separately as shown above. Then, we can multiply them using the property above to get e^((2+4)+(3-2)i) = e^(6+i).

In summary, to calculate the exponential of a complex number, use Euler's formula and the properties of exponents.