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Transcendental function From Wikipedia, the free encyclopedia

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Transcendental functionFrom Wikipedia, the free encyclopedia

Contents

1 nth root 11.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Origin of the root symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Etymology of “surd” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Definition and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Cube roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Identities and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Simplified form of a radical expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 Computing principal roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.7.1 nth root algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7.2 Digit-by-digit calculation of principal roots of decimal (base 10) numbers . . . . . . . . . 91.7.3 Logarithmic computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.8 Geometric constructibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.9 Complex roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.9.1 Square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.9.2 Roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.9.3 nth roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.10 Solving polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.13 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Transcendental function 152.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Algebraic and transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Transcendentally transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6 Exceptional set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.7 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

i

ii CONTENTS

2.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.11 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.11.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.11.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.11.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Chapter 1

nth root

Roots of integer numbers from 0 to 10. Line labels = x. x-axis = n. y-axis = nth root of x.

In mathematics, the nth root of a number x, where n is a positive integer, is a number r which, when raised to thepower n yields x

rn = x,

where n is the degree of the root. A root of degree 2 is called a square root and a root of degree 3, a cube root. Rootsof higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc.For example:

1

2 CHAPTER 1. NTH ROOT

• 2 is a square root of 4, since 22 = 4.

• −2 is also a square root of 4, since (−2)2 = 4.

A real number or complex number has n roots of degree n. While the roots of 0 are not distinct (all equaling 0), then nth roots of any other real or complex number are all distinct. If n is even and x is real and positive, one of its nthroots is positive, one is negative, and the rest are complex but not real; if n is even and x is real and negative, none ofthe nth roots is real. If n is odd and x is real, one nth root is real and has the same sign as x , while the other roots arenot real. Finally, if x is not real, then none of its nth roots is real.Roots are usually written using the radical symbol or radix√ or √ , with √

x or √x denoting the square root, 3√x

denoting the cube root, 4√x denoting the fourth root, and so on. In the expression n

√x , n is called the index, √ is

the radical sign or radix, and x is called the radicand. Since the radical symbol denotes a function, when a numberis presented under the radical symbol it must return only one result, so a non-negative real root, called the principalnth root, is preferred rather than others; if the only real root is negative, as for the cube root of –8, again the real rootis considered the principal root. An unresolved root, especially one using the radical symbol, is often referred to as asurd[1] or a radical.[2] Any expression containing a radical, whether it is a square root, a cube root, or a higher root,is called a radical expression, and if it contains no transcendental functions or transcendental numbers it is called analgebraic expression.In calculus, roots are treated as special cases of exponentiation, where the exponent is a fraction:

n√x = x1/n

Roots are particularly important in the theory of infinite series; the root test determines the radius of convergence ofa power series. Nth roots can also be defined for complex numbers, and the complex roots of 1 (the roots of unity)play an important role in higher mathematics. Galois theory can be used to determine which algebraic numbers canbe expressed using roots, and to prove the Abel-Ruffini theorem, which states that a general polynomial equation ofdegree five or higher cannot be solved using roots alone; this result is also known as “the insolubility of the quintic”.

1.1 Etymology

1.1.1 Origin of the root symbol

The origin of the root symbol √ is largely speculative. Some sources imply that the symbol was first used by Arabicmathematicians. One of those mathematicians was Abū al-Hasan ibn Alī al-Qalasādī (1421–1486). Legend has itthat it was taken from the Arabic letter " ج" (ǧīm, /dʒim/), which is the first letter in the Arabic word " جذر" (jadhir,meaning “root"; /ˈdʒɑːðir/).[3] However, many scholars, including Leonhard Euler,[4] believe it originates from theletter “r”, the first letter of the Latin word "radix" (meaning “root”), referring to the same mathematical operation.The symbol was first seen in print without the vinculum (the horizontal “bar” over the numbers inside the radicalsymbol) in the year 1525 in Die Coss by Christoff Rudolff, a German mathematician.The Unicode and HTML character codes for the radical symbols are:

1.1.2 Etymology of “surd”

The term surd traces back to al-Khwārizmī (c. 825), who referred to rational and irrational numbers as audible andinaudible, respectively. This later led to the Arabic word " أصم" (asamm, meaning “deaf” or “dumb”) for irrationalnumber being translated into Latin as “surdus” (meaning “deaf” or “mute”). Gerard of Cremona (c. 1150), Fibonacci(1202), and then Robert Recorde (1551) all used the term to refer to unresolved irrational roots.[5]

1.2 History

Main articles: Square root § History and Cube root § History

1.3. DEFINITION AND NOTATION 3

1.3 Definition and notation

+i

−i

−1 +1

The four 4th roots of −1,none of which is real

An nth root of a number x, where n is a positive integer, is any of the n real or complex numbers r whose nth poweris x:

rn = x.

Every positive real number x has a single positive nth root, called the principal nth root, which is written n√x . For

n equal to 2 this is called the principal square root and the n is omitted. The nth root can also be represented usingexponentiation as x1/n.For even values of n, positive numbers also have a negative nth root, while negative numbers do not have a real nthroot. For odd values of n, every negative number x has a real negative nth root. For example, −2 has a real 5th root,5√−2 = −1.148698354 . . . but −2 does not have any real 6th roots.

Every non-zero number x, real or complex, has n different complex number nth roots including any positive or negativeroots. They are all distinct except in the case of x = 0, all of whose nth roots equal 0.The nth roots of almost all numbers (all integers except the nth powers, and all rationals except the quotients of two

4 CHAPTER 1. NTH ROOT

0

+i

−i

−1 +1

The three 3rd roots of −1,one of which is a negative real

nth powers) are irrational. For example,

√2 = 1.414213562 . . .

All nth roots of integers, are algebraic numbers.

1.3.1 Square roots

Main article: Square root

A square root of a number x is a number r which, when squared, becomes x:

r2 = x.

Every positive real number has two square roots, one positive and one negative. For example, the two square roots of

1.3. DEFINITION AND NOTATION 5

0 2 4 6 8 10

1

2

3

−1

−2

−3

The graph y = ±√x .

25 are 5 and −5. The positive square root is also known as the principal square root, and is denoted with a radicalsign:

√25 = 5.

Since the square of every real number is a positive real number, negative numbers do not have real square roots.However, every negative number has two imaginary square roots. For example, the square roots of −25 are 5i and−5i, where i represents a square root of −1.

6 CHAPTER 1. NTH ROOT

0 2 4 6−2−4−6

1

2

−1

−2

The graph y = 3√x .

1.3.2 Cube roots

Main article: Cube root

A cube root of a number x is a number r whose cube is x:

r3 = x.

Every real number x has exactly one real cube root, written 3√x . For example,

3√8 = 2 and 3

√−8 = −2.

Every real number has two additional complex cube roots.

1.4 Identities and properties

Every positive real number has a positive nth root and the rules for operations with such surds are straightforward:

n√ab = n

√a

n√b ,

1.5. SIMPLIFIED FORM OF A RADICAL EXPRESSION 7

n

√a

b=

n√a

n√b.

Using the exponent form as in x1/n normally makes it easier to cancel out powers and roots.

n√am = (am)

1n = a

mn .

Problems can occur when taking the nth roots of negative or complex numbers. For instance:

√−1×

√−1 = −1

whereas

√(−1)× (−1) = 1

when taking the principal value of the roots.

1.5 Simplified form of a radical expression

A non-nested radical expression is said to be in simplified form if[6]

1. There is no factor of the radicand that can be written as a power greater than or equal to the index.

2. There are no fractions under the radical sign.

3. There are no radicals in the denominator.

For example, to write the radical expression√

325 in simplified form, we can proceed as follows. First, look for a

perfect square under the square root sign and remove it:

√325 =

√16·25 = 4

√25

Next, there is a fraction under the radical sign, which we change as follows:

4√

25 =

4√2√5

Finally, we remove the radical from the denominator as follows:

4√2√5

=4√2√5

·√5√5=

4√10

5=

4

5

√10

When there is a denominator involving surds it is always possible to find a factor to multiply both numerator anddenominator by to simplify the expression.[7][8] For instance using the factorization of the sum of two cubes:

13√a+ 3

√b=

3√a2 − 3

√ab+

3√b2

( 3√a+ 3

√b)(

3√a2 − 3

√ab+

3√b2)

=3√a2 − 3

√ab+

3√b2

a+ b.

Simplifying radical expressions involving nested radicals can be quite difficult. It is not immediately obvious forinstance that:

√3 + 2

√2 = 1 +

√2

8 CHAPTER 1. NTH ROOT

1.6 Infinite series

The radical or root may be represented by the infinite series:

(1 + x)s/t =

∞∑n=0

∏n−1k=0(s− kt)

n!tnxn

with |x| < 1 . This expression can be derived from the binomial series.

1.7 Computing principal roots

The nth root of an integer is not always an integer, and if it is not an integer then it is not a rational number. Forinstance, the fifth root of 34 is

5√34 = 2.024397458 . . . ,

where the dots signify that the decimal expression does not end after any finite number of digits. Since in this examplethe digits after the decimal never enter a repeating pattern, the number is irrational.

1.7.1 nth root algorithm

The nth root of a number A can be computed by the nth root algorithm, a special case of Newton’s method. Startwith an initial guess x0 and then iterate using the recurrence relation

xk+1 =1

n

((n− 1)xk +

A

xn−1k

)until the desired precision is reached.Depending on the application, it may be enough to use only the first Newton approximant:

n√xn + y ≈ x+

y

nxn−1.

For example, to find the fifth root of 34, note that 25 = 32 and thus take x = 2, n = 5 and y = 2 in the above formula.This yields

5√34 = 5

√32 + 2 ≈ 2 +

2

5 · 16= 2.025.

The error in the approximation is only about 0.03%.Newton’s method can be modified to produce a generalized continued fraction for the nth root which can be modifiedin various ways as described in that article. For example:

n√z = n

√xn + y = x+

y

nxn−1 +(n− 1)y

2x+(n+ 1)y

3nxn−1 +(2n− 1)y

2x+(2n+ 1)y

5nxn−1 +(3n− 1)y

2x+. . .

;

1.7. COMPUTING PRINCIPAL ROOTS 9

n√z = x+

2x · y

n(2z − y)− y −(12n2 − 1)y2

3n(2z − y)−(22n2 − 1)y2

5n(2z − y)−(32n2 − 1)y2

7n(2z − y)−. . .

.

In the case of the fifth root of 34 above (after dividing out selected common factors):

5√34 = 2 +

1

40 +4

4 +6

120 +9

4 +11

200 +14

4 +. . .

= 2 +4 · 1

165− 1−4 · 6

495−9 · 11

825−14 · 16

1155−. . .

.

1.7.2 Digit-by-digit calculation of principal roots of decimal (base 10) numbers

Pascal’s Triangle showing P (4, 1) = 4 .

Building on the digit-by-digit calculation of a square root, it can be seen that the formula used there, x(20p+x) ≤ c, or x2 + 20xp ≤ c , follows a pattern involving Pascal’s triangle. For the nth root of a number P (n, i) is definedas the value of element i in row n of Pascal’s Triangle such that P (4, 1) = 4 , we can rewrite the expression as∑n−1

i=0 10iP (n, i)pixn−i . For convenience, call the result of this expression y . Using this more general expression,any positive principal root can be computed, digit-by-digit, as follows.Write the original number in decimal form. The numbers are written similar to the long division algorithm, and, asin long division, the root will be written on the line above. Now separate the digits into groups of digits equating tothe root being taken, starting from the decimal point and going both left and right. The decimal point of the root willbe above the decimal point of the square. One digit of the root will appear above each group of digits of the originalnumber.Beginning with the left-most group of digits, do the following procedure for each group:

1. Starting on the left, bring down the most significant (leftmost) group of digits not yet used (if all the digitshave been used, write “0” the number of times required to make a group) and write them to the right of theremainder from the previous step (on the first step, there will be no remainder). In other words, multiply theremainder by 10n and add the digits from the next group. This will be the current value c.

10 CHAPTER 1. NTH ROOT

2. Find p and x, as follows:

• Let p be the part of the root found so far, ignoring any decimal point. (For the first step, p = 0 ).• Determine the greatest digit x such that y ≤ c .• Place the digit x as the next digit of the root, i.e., above the group of digits you just brought down. Thus

the next p will be the old p times 10 plus x.

3. Subtract y from c to form a new remainder.

4. If the remainder is zero and there are no more digits to bring down, then the algorithm has terminated. Oth-erwise go back to step 1 for another iteration.

Examples

Find the square root of 152.2756.1 2. 3 4 / \/ 01 52.27 56 01 100·1·00·12 + 101·2·01·11 ≤ 1 < 100·1·00·22 + 101·2·01·21 x = 1 01 y = 100·1·00·12 +101·2·01·12 = 1 + 0 = 1 00 52 100·1·10·22 + 101·2·11·21 ≤ 52 < 100·1·10·32 + 101·2·11·31 x = 2 00 44 y = 100·1·10·22

+ 101·2·11·21 = 4 + 40 = 44 08 27 100·1·120·32 + 101·2·121·31 ≤ 827 < 100·1·120·42 + 101·2·121·41 x = 3 07 29y = 100·1·120·32 + 101·2·121·31 = 9 + 720 = 729 98 56 100·1·1230·42 + 101·2·1231·41 ≤ 9856 < 100·1·1230·52

+ 101·2·1231·51 x = 4 98 56 y = 100·1·1230·42 + 101·2·1231·41 = 16 + 9840 = 9856 00 00 Algorithm terminates:Answer is 12.34Find the cube root of 4192 to the nearest hundredth.1 6. 1 2 4 3 / \/ 004 192.000 000 000 004 100·1·00·13 + 101·3·01·12 + 102·3·02·11 ≤ 4 < 100·1·00·23 + 101·3·01·22 +102·3·02·21 x = 1 001 y = 100·1·00·13 + 101·3·01·12 + 102·3·02·11 = 1 + 0 + 0 = 1 003 192 100·1·10·63 + 101·3·11·62

+ 102·3·12·61 ≤ 3192 < 100·1·10·73 + 101·3·11·72 + 102·3·12·71 x = 6 003 096 y = 100·1·10·63 + 101·3·11·62

+ 102·3·12·61 = 216 + 1,080 + 1,800 = 3,096 096 000 100·1·160·13 + 101·3·161·12 + 102·3·162·11 ≤ 96000 <100·1·160·23 + 101·3·161·22 + 102·3·162·21 x = 1 077 281 y = 100·1·160·13 + 101·3·161·12 + 102·3·162·11 = 1 +480 + 76,800 = 77,281 018 719 000 100·1·1610·23 + 101·3·1611·22 + 102·3·1612·21 ≤ 18719000 < 100·1·1610·33 +101·3·1611·32 + 102·3·1612·31 x = 2 015 571 928 y = 100·1·1610·23 + 101·3·1611·22 + 102·3·1612·21 = 8 + 19,320+ 15,552,600 = 15,571,928 003 147 072 000 100·1·16120·43 + 101·3·16121·42 + 102·3·16122·41 ≤ 3147072000 <100·1·16120·53 + 101·3·16121·52 + 102·3·16122·51 x = 4 The desired precision is achieved: The cube root of 4192is about 16.12

1.7.3 Logarithmic computation

The principal nth root of a positive number can be computed using logarithms. Starting from the equation thatdefines r as an nth root of x, namely rn = x, with x positive and therefore its principal root r also positive, one takeslogarithms of both sides (any base of the logarithm will do; base 10 is used here) to obtain

n log10 r = log10 x hence log10 r =log10 x

n.

The root r is recovered from this by taking the antilog:

r = 10log10 x

n .

For the case in which x is negative and n is odd, there is one real root r which is also negative. This can be found byfirst multiplying both sides of the defining equation by –1 to obtain |r|n = |x|, then proceeding as before to find |r|,and using r = –|r|.

1.8 Geometric constructibility

The ancient Greek mathematicians knew how to use compass and straightedge to construct a length equal to thesquare root of a given length. In 1837 Pierre Wantzel proved that an nth root of a given length cannot be constructedif n > 2.

1.9. COMPLEX ROOTS 11

1.9 Complex roots

Every complex number other than 0 has n different nth roots.

1.9.1 Square roots

0

+i

−i

−1 +1

The square roots of i

The two square roots of a complex number are always negatives of each other. For example, the square roots of −4are 2i and −2i, and the square roots of i are

1√2(1 + i) and − 1√

2(1 + i).

If we express a complex number in polar form, then the square root can be obtained by taking the square root of theradius and halving the angle:

√reiθ = ±

√r eiθ/2.

12 CHAPTER 1. NTH ROOT

A principal root of a complex number may be chosen in various ways, for example

√reiθ =

√r eiθ/2

which introduces a branch cut in the complex plane along the positive real axis with the condition 0 ≤ θ < 2π, oralong the negative real axis with −π < θ ≤ π.Using the first(last) branch cut the principal square root √z maps z to the half plane with non-negative imaginary(real)part. The last branch cut is presupposed in mathematical software like Matlab or Scilab.

1.9.2 Roots of unity

=

The three 3rd roots of 1

Main article: Root of unity

The number 1 has n different nth roots in the complex plane, namely

1, ω, ω2, . . . , ωn−1,

where

ω = e2πi/n = cos(2π

n

)+ i sin

(2π

n

)

1.10. SOLVING POLYNOMIALS 13

These roots are evenly spaced around the unit circle in the complex plane, at angles which are multiples of 2π/n .For example, the square roots of unity are 1 and −1, and the fourth roots of unity are 1, i , −1, and −i .

1.9.3 nth roots

Every complex number has n different nth roots in the complex plane. These are

η, ηω, ηω2, . . . , ηωn−1,

where η is a single nth root, and 1, ω, ω2, ... ωn−1 are the nth roots of unity. For example, the four different fourthroots of 2 are

4√2, i

4√2, − 4

√2, and − i

4√2.

In polar form, a single nth root may be found by the formula

n√reiθ = n

√r eiθ/n.

Here r is the magnitude (the modulus, also called the absolute value) of the number whose root is to be taken; ifthe number can be written as a+bi then r =

√a2 + b2 . Also, θ is the angle formed as one pivots on the origin

counterclockwise from the positive horizontal axis to a ray going from the origin to the number; it has the propertiesthat cos θ = a/r, sin θ = b/r, and tan θ = b/a.

Thus finding nth roots in the complex plane can be segmented into two steps. First, the magnitude of all the nth rootsis the nth root of the magnitude of the original number. Second, the angle between the positive horizontal axis anda ray from the origin to one of the nth roots is θ/n , where θ is the angle defined in the same way for the numberwhose root is being taken. Furthermore, all n of the nth roots are at equally spaced angles from each other.If n is even, a complex number’s nth roots, of which there are an even number, come in additive inverse pairs, so thatif a number r1 is one of the nth roots then r2 = –r1 is another. This is because raising the latter’s coefficient –1 to thenth power for even n yields 1: that is, (–r1)n = (–1)n × r1n = r1n.As with square roots, the formula above does not define a continuous function over the entire complex plane, butinstead has a branch cut at points where θ / n is discontinuous.

1.10 Solving polynomials

See also: Root-finding algorithm

It was once conjectured that all polynomial equations could be solved algebraically (that is, that all roots of a polynomialcould be expressed in terms of a finite number of radicals and elementary operations). However, while this is true forthird degree polynomials (cubics) and fourth degree polynomials (quartics), the Abel-Ruffini theorem (1824) showsthat this is not true in general when the degree is 5 or greater. For example, the solutions of the equation

x5 = x+ 1

cannot be expressed in terms of radicals. (cf. quintic equation)

1.11 See also• Nth root algorithm

• Shifting nth-root algorithm

14 CHAPTER 1. NTH ROOT

• Irrational number

• Algebraic number

• Nested radical

• Twelfth root of two

• Super-root

1.12 References[1] Bansal, R K (2006). New Approach to CBSE Mathematics IX. Laxmi Publications. p. 25. ISBN 978-81-318-0013-3.

[2] Silver, Howard A. (1986). Algebra and trigonometry. Englewood Cliffs, N.J.: Prentice-Hall. ISBN 0-13-021270-9.

[3] “Language Log: Ab surd". Retrieved 22 June 2012.

[4] Leonhard Euler (1755). Institutiones calculi differentialis (in Latin).

[5] “Earliest Known Uses of Some of the Words of Mathematics”. Mathematics Pages by Jeff Miller. Retrieved 2008-11-30.

[6] McKeague, Charles P. (2011). Elementary algebra. p. 470.

[7] B.F. Caviness, R.J. Fateman, “Simplification of Radical Expressions”, Proceedings of the 1976ACMSymposium on Symbolicand Algebraic Computation, p. 329 full text

[8] Richard Zippel, “Simplification of Expressions Involving Radicals”, Journal of Symbolic Computation 1:189-210 (1985)doi:10.1016/S0747-7171(85)80014-6

1.13 External links

Chapter 2

Transcendental function

A transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to analgebraic function.[1][2] (The polynomials are sometimes required to have rational coefficients.) In other words, atranscendental function “transcends” algebra in that it cannot be expressed in terms of a finite sequence of thealgebraic operations of addition, multiplication, and root extraction.Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions.

2.1 Definition

Formally, an analytic function ƒ(z) of one real or complex variable z is transcendental if it is algebraically independentof that variable.[3] This can be extended to functions of several variables.

2.2 History

Transcendental functions were first defined by Leonhard Euler in his Introductio (1748) as functions either not defin-able by the “ordinary operations of algebra”, or defined by such operations “repeated infinitely often”. But this defini-tion is unsatisfactory, since some functions defined with infinitely many operations remain algebraic or even rational.The theory was further developed by Gotthold Eisenstein (Eisenstein’s theorem), Eduard Heine, and others.[4]

2.3 Examples

The following functions are transcendental:

f1(x) = xπ

f2(x) = cx, c ̸= 0, 1

f3(x) = xx

f4(x) = x1x

f5(x) = logc x, c ̸= 0, 1

f6(x) = sinx

In particular, for ƒ2 if we set c equal to e, the base of the natural logarithm, then we get that ex is a transcendentalfunction. Similarly, if we set c equal to e in ƒ5, then we get that ln(x), the natural logarithm, is a transcendentalfunction.

15

16 CHAPTER 2. TRANSCENDENTAL FUNCTION

2.4 Algebraic and transcendental functions

For more details on this topic, see Elementary function (differential algebra).

The most familiar transcendental functions are the logarithm, the exponential (with any non-trivial base), the trigonometric,and the hyperbolic functions, and the inverses of all of these. Less familiar are the special functions of analysis, suchas the gamma, elliptic, and zeta functions, all of which are transcendental. The generalized hypergeometric andBessel functions are transcendental in general, but algebraic for some special parameter values.A function that is not transcendental is algebraic. Simple examples of algebraic functions are the rational functionsand the square root function, but in general, algebraic functions cannot be defined as finite formulas of the elementaryfunctions.[5]

The indefinite integral of many algebraic functions is transcendental. For example, the logarithm function arose fromthe reciprocal function in an effort to find the area of a hyperbolic sector.Differential algebra examines how integration frequently creates functions that are algebraically independent of someclass, such as when one takes polynomials with trigonometric functions as variables.

2.5 Transcendentally transcendental functions

Most of the familiar transcendental functions, including the special functions of mathematical physics, are solutionsof algebraic differential equations. Those which are not, such as the gamma and the zeta functions, are called tran-scendentally transcendental or hypertranscendental functions.

2.6 Exceptional set

If ƒ(z) is an algebraic function and α is an algebraic number then ƒ(α) will also be an algebraic number. The converseis not true: there are entire transcendental functions ƒ(z) such that ƒ(α) is an algebraic number for any algebraic α.[6]

In many instances, however, the set of algebraic numbers α where ƒ(α) is algebraic is fairly small. For example, if ƒ isthe exponential function, ƒ(z) = ez, then the only algebraic number α where ƒ(α) is also algebraic is α = 0, where ƒ(α)= 1. For a given transcendental function this set of algebraic numbers giving algebraic results is called the exceptionalset of the function,[7][8] that is the set

E(f) = {α ∈ Q : f(α) ∈ Q}.

If this set can be calculated then it can often lead to results in transcendental number theory. For example, Lindemannproved in 1882 that the exceptional set of the exponential function is just {0}. In particular exp(1) = e is transcen-dental. Also, since exp(iπ) = −1 is algebraic we know that iπ cannot be algebraic. Since i is algebraic this impliesthat π is a transcendental number.In general, finding the exceptional set of a function is a difficult problem, but it has been calculated for some functions:

• E(exp) = {0} ,

• E(j) = {α ∈ H : [Q(α) : Q] = 2} ,

• Here j is Klein’s j-invariant, H is the upper half-plane, and [Q(α): Q] is the degree of the number fieldQ(α). This result is due to Theodor Schneider.[9]

• E(2x) = Q ,

• This result is a corollary of the Gelfond–Schneider theorem which says that if α is algebraic and not 0 or1, and if β is algebraic and irrational then αβ is transcendental. Thus the function 2x could be replacedby cx for any algebraic c not equal to 0 or 1. Indeed, we have:

2.7. DIMENSIONAL ANALYSIS 17

• E(xx) = E(x 1x ) = Q \ {0}.

• A consequence of Schanuel’s conjecture in transcendental number theory would be that E(eex) = ∅.

• A function with empty exceptional set that does not require assuming Schanuel’s conjecture is ƒ(x) = exp(1 +πx).

While calculating the exceptional set for a given function is not easy, it is known that given any subset of the algebraicnumbers, say A, there is a transcendental function ƒ whose exceptional set is A.[10] The subset does not need to beproper, meaning that A can be the set of algebraic numbers. This directly implies that there exist transcendentalfunctions that produce transcendental numbers only when given transcendental numbers. Alex Wilkie also provedthat there exist transcendental functions for which first-order-logic proofs about their transcendence do not exist byproviding an exemplary analytic function.[11]

2.7 Dimensional analysis

In dimensional analysis, transcendental functions are notable because they make sense only when their argument isdimensionless (possibly after algebraic reduction). Because of this, transcendental functions can be an easy-to-spotsource of dimensional errors. For example, log(5 meters) is a nonsensical expression, unlike log(5 meters / 3 meters)or log(3) meters. One could attempt to apply a logarithmic identity to get log(10) + log(m), which highlights theproblem: applying a non-algebraic operation to a dimension creates meaningless results.

2.8 See also• Complex function• Function (mathematics)• Generalized function• List of special functions and eponyms• List of types of functions• Rational function• Special functions

2.9 References[1] E. J. Townsend, Functions of a Complex Variable, 1915, p. 300

[2] Michiel Hazewinkel, Encyclopedia of Mathematics, 1993, 9:236

[3] M. Waldschmidt, Diophantine approximation on linear algebraic groups, Springer (2000).

[4] Amy Dahan-Dalmédico, Jeanne Peiffer, History of Mathematics: Highways and Byways, 2010, p. 240

[5] cf. Abel–Ruffini theorem

[6] A. J. van der Poorten. 'Transcendental entire functions mapping every algebraic number field into itself’, J. Austral. Math.Soc. 8 (1968), 192–198

[7] D. Marques, F. M. S. Lima, Some transcendental functions that yield transcendental values for every algebraic entry, (2010)arXiv:1004.1668v1.

[8] N. Archinard, Exceptional sets of hypergeometric series, Journal of Number Theory 101 Issue 2 (2003), pp.244–269.

[9] T. Schneider, Arithmetische Untersuchungen elliptischer Integrale, Math. Annalen 113 (1937), pp.1–13.

[10] M. Waldschmidt, Auxiliary functions in transcendental number theory, The Ramanujan Journal 20 no3, (2009), pp.341–373.

[11] A. Wilkie, An algebraically conservative, transcendental function, Paris VII preprints, number 66, 1998.

18 CHAPTER 2. TRANSCENDENTAL FUNCTION

2.10 External links• Definition of “Transcendental function” in the Encyclopedia of Math

2.11. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 19

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2.11.1 Text• Nth root Source: https://en.wikipedia.org/wiki/Nth_root?oldid=681133170Contributors: AxelBoldt, Zundark, SimonP, Agitate, Nealmcb,

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