23 junio, 2024

Transcendent functions: types, definition, properties, examples

The transcendental functions elementals are lace exponentials, llogarithmic ace, ltrigonometric ace, lthe inverse trigonometric functions, lhyperbolic ace and linverse hyperbolic ace. That is, they are those that cannot be expressed by a polynomial, a quotient of polynomials, or roots of polynomials.

Non-elementary transcendental functions are also known as special functions and among them the error function can be named. The algebraic functions (polynomials, quotients of polynomials, and roots of polynomials) next to the transcendental functions elements constitute what in mathematics is known as elementary functions.

Transcendental functions are also considered those that result from operations between transcendental functions or between transcendental and algebraic functions. These operations are: the sum and difference of functions, product and quotient of functions, as well as the composition of two or more functions.

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Definition and properties

Exponential function

It is a real function of a real independent variable of the form:

f(x) = a^x = ax

where to is a positive real number (a>0) fixed called the base. Circumflex or superscript are used to denote the exponentiation operation.

Let’s say that a = 2 then the function looks like this:

f(x) = 2^x = 2x

Which will be evaluated for various values ​​of the independent variable x:

Below is a graph where the exponential function is represented for various values ​​of the base, including the base and (number of Neper and ≃ 2.72). Base and is so important that, generally, when talking about an exponential function, one thinks of e^xwhich is also denoted exp(x).

Properties of the exponential function

From figure 1 it can be seen that the domain of exponential functions are the real numbers (Dom f = R.) and the range or route are the positive reals (Ran f = R.+).

On the other hand, regardless of the value of the base a, all exponential functions pass through the point (0, 1) and through the point (1, a).

when the base to > 1then the function is increasing and when 0 < to < 1 the function is decreasing.

The curves of y=a^x and of y= (1/a)^x They are symmetric about the axis. AND.

With the exception of the case a=1the exponential function is injective, that is to say, each value of the image corresponds to one and only one starting value.

logarithmic function

It is a real function of real independent variable based on the definition of the logarithm of a number. The logarithm to the base to of a number xIt’s the number and to which the base must be raised to obtain the argument x:

logto(x) = y ⇔ a^y = x

That is to say, that the logarithm function in base to is the inverse function to the exponential function in base to.

For example:

log21 = 0, since 2^0 =1

Another case, log24 = 2, because 2^2 =4

The root logarithm of 2 is log2√2 = ½ , since 2^½ =√2

log2 ¼ = -2, since 2^(-2) = ¼

Below is a graph of the logarithm function in various bases.

Properties of the logarithm function

The domain of the logarithm function y(x) = logto(x) are the positive real numbers R+. The range or tour are the real numbers R..

Regardless of the base, the logarithm function always passes through the point (1,0) and the point (a, 1) belongs to the graph of said function.

In the event that the base a is greater than unity (a > 1) the logarithm function is increasing. But if (0 < a < 1) then it is a decreasing function.

sine, cosine, and tangent functions

The sine function assigns a real number y to each x-value, where x represents the measure of an angle in radians. To obtain the value of Sin(x) of an angle, the angle is represented on the unit circle and the projection of said angle on the vertical axis is the sine corresponding to that angle.

Below is shown (in figure 3) the trigonometric circle and the sine for various angular values ​​X1, X2, X3 and X4.

Defined in this way, the maximum value that the function Sin(x) can have is 1, which occurs when x= π/2 + 2π n, where n is an integer (0,±1, ±2, ). The minimum value that the function Sin(x) can take occurs when x = 3π/2 + 2π n.

The function cosine y = Cos(x) is defined in a similar way, but the projection of the angular positions P1, P2, etc. is done on the horizontal axis of the trigonometric circle.

On the other hand, the function y = Tan(x) is the quotient between the sine function and the cosine function.

Below is a graph of the transcendental functions Sin(x), Cos(x) and Tan(x)

Derivatives and integrals

Derivative of the exponential function

the derivative and’ of the exponential function y = a^x is the function a^x multiplied by the natural logarithm of the base a:

y’ = (a^x)’ = a^x ln a

In the particular case of the basis andthe derivative of the exponential function is the exponential function itself.

Integral of exponential function

The indefinite integral of a^x is the function itself divided by the natural logarithm of the base.

In the particular case of the base e, the integral of the exponential function is the exponential function itself.

Table of derivatives and integrals of transcendental functions

Below is a summary table of the main transcendental functions, their derivatives and indefinite integrals (antiderivatives):

examples

Example 1

Find the function resulting from the composition of the function f(x) = x^3 with the function g(x) = cos(x):

(fog) (x) = f(g(x)) = cos3(x)

Its derivative and its indefinite integral is:

Example 2

Find the composition of the function g with the function f, being g and f the functions defined in the previous example:

(gof) (x) = g(f(x)) = cos(x3)

It should be noted that the composition of functions is not a commutative operation.

The derivative and the indefinite integral for this function are respectively:

The integral was left indicated because it is not possible to write the result as a combination of elementary functions in exact form.

References

Calculus of a Single Variable. Ron Larson, Bruce H. Edwards. Cengage Learning, 10 Nov. 2008
The Implicit Function Theorem: History, Theory, and Applications. Steven G. Krantz, Harold R. Parks. Springer Science & Business Media, 9 Nov. 2012
Multivariate Analysis. Satish Shirali, Harkrishan Lal Vasudeva. Springer Science & Business Media, Dec. 13. 2010
System Dynamics: Modeling, Simulation, and Control of Mechatronic Systems. Dean C. Karnopp, Donald L. Margolis, Ronald C. Rosenberg. John Wiley & Sons, 7 Mar. 2012
Calculus: Mathematics and Modeling. William Bauldry, Joseph R. Fiedler, Frank R. Giordano, Ed Lodi, Rick Vitray. Addison Wesley Longman, Jan. 1 1999
wikipedia. transcendent function. Recovered from: en.wikipedia.com

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