Exponential function

In mathematics, an exponential function is a function of the formz = Re(ex + iy)z = Im(ex + iy)z = |ex + iy|Checker board key: x > 0 : green {displaystyle x>0:;{ ext{green}}} x < 0 : red {displaystyle x<0:;{ ext{red}}} y > 0 : yellow {displaystyle y>0:;{ ext{yellow}}} y < 0 : blue {displaystyle y<0:;{ ext{blue}}} Projection onto the range complex plane (V/W). Compare to the next, perspective picture.Projection into the x {displaystyle x} , v {displaystyle v} , and w {displaystyle w} dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image).Projection into the y {displaystyle y} , v {displaystyle v} , and w {displaystyle w} dimensions, producing a spiral shape. ( y {displaystyle y} range extended to ±2π, again as 2-D perspective image). In mathematics, an exponential function is a function of the form where b is a positive real number, and in which the argument x occurs as an exponent. For real numbers c and d, a function of the form f ( x ) = a b c x + d {displaystyle f(x)=ab^{cx+d}} is also an exponential function, as it can be rewritten as As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (that is, its derivative) is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b: For b = 1 the real exponential function is a constant and the derivative is zero because log e ⁡ b = 0 , {displaystyle log _{e}b=0,} for positive a and b > 1 the real exponential functions are monotonically increasing (as depicted for b = e and b = 2), because the derivative is greater than zero for all arguments, and for b < 1 they are monotonically decreasing (as depicted for b = 1/2), because the derivative is less than zero for all arguments. The constant e = 2.71828... is the unique base for which the constant of proportionality is 1, so that the function's derivative is itself: Since changing the base of the exponential function merely results in the appearance of an additional constant factor, it is computationally convenient to reduce the study of exponential functions in mathematical analysis to the study of this particular function, conventionally called the 'natural exponential function', or simply, 'the exponential function' and denoted by