LOG#023. Math survey.

What is a triangle? It is a question of definition in Mathematics. Of course you could disagree, but it is true. Look the above three “triangles”. Euclidean geometry is based in the first one. The second “triangle” is commonly found in special relativity. Specially, hyperbolic functions. The third one is related to spherical/elliptical geometry.

Today’s summary: some basic concepts in arithmetics, complex numbers and functions. We are going to study and review the properties of some elementary and well known functions. We are doing this in order to prepare a better background for the upcoming posts, in which some special functions will appear. Maybe, this post can be useful for understanding some previous posts too.

First of all, let me remember you that elementary arithmetics is based on seven basic “operations”: addition, substraction, multiplication, division, powers, roots, exponentials and logarithms. You are familiar with the 4 first operations, likely you will also know about powers and roots, but exponentials and logarithms are the last kind of elementary operations taught in the school ( high school, in the case they are ever explained!).

Let me begin with addition/substraction of real numbers (it would be also valid for complex numbers $z=a+bi$ or even more general “numbers”, “algebras”, “rings” or “fields”, with suitable extensions).

$a+b=b+a$

$(a+b)+c=a+(b+c)$

$a+(-a)=0$

$a+0=a$

Multiplication is a harder operation. We have to be careful with the axioms since there are many places in physics where multiplication is generaliz loosing some of the following properties:

$kA=\underbrace{A+..+A}_\text{k-times}$

$AB=BA$

$(AB)C=A(BC)=ABC$

$1A=A1=A$

$(A+B)C=AC+BC$

$A(B+C)=AB+AC$

$A^{-1}A=AA^{-1}=1$

Indeed the last rule can be undestood as the “division” rule, provided $A\neq 0$ since in mathematics or physics there is no sense to “divide by zero”, as follows.

$A^{-1}=\dfrac{1}{A}$

Now, we are going to review powers and roots.

$x^a=\underbrace{x\cdots x}_\text{a-times}$

$(x^a)^b=x^{ab}$

$x^{-a}=\dfrac{1}{x^a}$

$x^ax^b=x^{a+b}$

$\sqrt[n]{x}=x^{1/n}$

Note that the identity $x^a+y^a=(x+y)^a$ is not true in general. Moreover, if $x\neq 0$ then $x^0=1$ as well, as it can be easily deduced from the previous axioms. Now, the sixth operation is called exponentiation. It reads:

$\exp (a+b)=\exp (a)\exp (b)$

Sometimes you can read $e^{a+b}=e^{a}e^{b}$ , where $\displaystyle{e=\lim_{x\to\infty}\left(a+\dfrac{1}{n}\right)^n}$ is the so-called “e” number. The definitions is even more general, since the previous property is the key feature for any exponential. I mean that,

$a^x=\underbrace{a\cdots a}_\text{x-times}$

$a^xb^x=(ab)^x$

$\left(\dfrac{a}{b}\right)^x=\dfrac{a^x}{b^x}$

We also get that for any $x\neq 0$ , then $0^x=0$ . Finally, the 7th operation. Likely, the most mysterious for the layman. However, it is very useful in many different places. Recall the definition of the logarithm in certain base “a”:

$\log_{(a)} x=y \leftrightarrow x=a^y$

Please, note that this definition has nothing to do with the “deformed” logarithm of my previous log-entry. Notations are subtle, but you must always be careful about what are you talking about!

Furthermore, there are more remarks:

1st. Sometimes you write $\log_e=\ln x$ . Be careful, some books use other notations for the Napier’s logarithm/natural logarithm. Then, you can find out there $\log_e=L$ or even $\log_e=\log$ .

2nd. Whenever you are using a calculator, you can generally find $\log_e=\ln$ and $\log_{(10)}=\log$ . Please, note that in this case $\log$ is not the natural logarithm, it is the decimal logarithm.

Logarithms (caution: logarithms of real numbers, since the logarithms of complex numbers are a bit more subtle) have some other cool properties:

$\log_{(a)}(xy)=\log_{(a)}x+\log_{(a)}y$

$\log_{(a)}\dfrac{x}{y}=\log_{(a)}x-\log_{(a)}y$

$\log_{(a)}x^y=y\log_{(a)}x$

$\log_{(a)}=\dfrac{\log_{(b)}}{\log_{(b)}a}$

Common values of the logarithm are:

$\ln 0^+=-\infty;\; \ln 1=0;\; \ln e=1;\; \ln e^x=e^{\ln x}=x$

Indeed, logarithms are also famous due to a remarkable formula by Dirac to express any number in terms of 2’s as follows:

$\displaystyle{N=-\log_2\log_2 \sqrt{\sqrt{\underbrace{\cdots}_\text{(N-1)-times}2}}=-\log_2\log_2\sqrt{\underbrace{\cdots}_\text{(N)-times}2}}$

However, it is quite a joke, since it is even easier to write $N=\log_a a^N$ , or even $N=\log_{(1/a)}a^{-N}$

Are we finished? NO! There are more interesting functions to review. In particular, the trigonometric functions are the most important functions you can find in the practical applications.

EUCLIDEAN TRIGONOMETRY

Triangles are cool! Let me draw the basic triangle in euclidean trigonometry.

The trigonometric ratios/functions you can define from this figure are:

i)The function (sin), defined as the ratio of the side opposite the angle to the hypotenuse:

$\sin A=\dfrac{\textrm{opposite}}{\textrm{hypotenuse}}=\dfrac{a}{\,c\,}$

ii) The function (cos), defined as the ratio of the adjacent leg to the hypotenuse.

$\cos A=\dfrac{\textrm{adjacent}}{\textrm{hypotenuse}}=\dfrac{b}{\,c\,}$

iii) The function (tan), defined as the ratio of the opposite leg to the adjacent leg.

$\tan A=\dfrac{\textrm{opposite}}{\textrm{adjacent}}=\dfrac{a}{\,b\,}=\dfrac{\sin A}{\cos A}$

The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle ”A”. The ”’adjacent leg”’ is the other side that is adjacent to angle ”A”. The ”’opposite side”’ is the side that is opposite to angle ”A”. The terms ”’perpendicular”’ and ”’base”’ are sometimes used for the opposite and adjacent sides respectively. Many English speakers find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA ( a mnemonics rule whose derivation and meaning is left to the reader).

The multiplicative inverse or reciprocals of these functions are named the cosecant (csc or cosec), secant(sec), and cotangent (cot), respectively:

$\csc A=\dfrac{1}{\sin A}=\dfrac{c}{a}$

$\sec A=\dfrac{1}{\cos A}=\dfrac{c}{b}$

$\cot A=\dfrac{1}{\tan A}=\dfrac{\cos A}{\sin A}=\dfrac{b}{a}$

The inverse trigonometric functions/inverse functions are called the arcsine, arccosine, and arctangent, respectively. These functions are what in common calculators are given by $\sin^{-1},\cos^{-1},\tan^{-1}$ . Don’t confuse them with the multiplicative inverse trigonometric functions.

There are arithmetic relations between these functions, which are known as trigonometric identities. The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to “co-“. From the goniometric circle (a circle of radius equal to 1) you can read the Fundamental Theorem of (euclidean) Trigonometry:

$\cos^2\theta+\sin^2\theta=1$

Indeed, from the triangle above, you can find out that the pythagorean theorem implies

$a^2+b^2=c^2$

or equivalently

$\dfrac{a^2}{c^2}+\dfrac{b^2}{c^2}=\dfrac{c^2}{c^2}$

So, the Fundamental Theorem of Trigonometry is just a dressed form of the pythagorean theorem!
The fundamental theorem of trigonometry can be rewritten too as follows:

$\tan^2\theta+1=\sec^2\theta$

$\cot^2+1=\csc^2\theta$

These equations can be easily derived geometrically from the goniometric circle:

The trigonometric ratios are also related geometrically to this circle, and it can seen from the next picture:

Other trigonometric identities are:

$\sin (x\pm y)=\sin x \cos y\pm \sin y\cos x$

$\cos (x\pm y)=\cos x \cos y -\sin x \sin y$

$\tan (x\pm y)=\dfrac{\tan x\pm \tan y}{1\mp \tan x \tan y}$

$\cot (x\pm y)=\dfrac{\cot x \cot y\mp 1}{\cot x\pm \cot y}$

$\sin 2x=2\sin x \cos x$

$\cos 2x=\cos^2 x-\sin^2 x$

$\tan 2x=\dfrac{2\tan x}{1-\tan^2 x}$

$\sin \dfrac{x}{2}=\sqrt{\dfrac{1-\cos x}{2}}$

$\cos \dfrac{x}{2}=\sqrt{\dfrac{1+\cos x}{2}}$

$\tan \dfrac{x}{2}=\sqrt{\dfrac{1-\cos x}{1+\cos x}}$

$\sin x\sin y=\dfrac{\cos(x-y)-\cos(x+y)}{2}$

$\sin x\cos y=\dfrac{\sin(x+y)+\sin(x-y)}{2}$

$\cos x\cos y=\dfrac{\cos (x+y)+\cos(x-y)}{2}$

The above trigonometric functions are also valid for complex numbers with care enough. Let us write a complex number as either a binomial expression $z=a+bi$ or like a trigonometric expression $z=re^{i\theta}$ . The famous Euler identity:

$e^{i\theta}=\cos \theta + i\sin \theta$

allows us to relate both two expressions for a complex number since

$z=r(\cos \theta + i\sin \theta)$

implies that $a=r\cos\theta$ and $b=r\sin\theta$ . The Euler formula is also useful to recover the identities for the sin and cos of a sum/difference, since $e^{iA}e^{iB}=e^{i(A+B)}$

The complex conjugate of a complex number is $\bar{z}=a-bi$ , and the modulus is

$z\bar{z}=\vert z\vert^2=r^2$

with $\theta =\arctan \dfrac{b}{a}, \;\; \vert z \vert=\sqrt{a^2+b^2}$

Moreover, $\overline{\left(z_1\pm z_2\right)}=\bar{z_1}\pm\bar{z_2}$ , $\vert \bar{z}\vert=\vert z\vert$ , and if $z_2\neq 0$ , then

$\overline{\left(\dfrac{z_1}{z_2}\right)}=\dfrac{\bar{z_1}}{\bar{z_2}}$

We also have the so-called Moivre’s formula

$z^n=r^n(\cos n\theta+i\sin n\theta)$

and for the complex roots of complex numbers with $w^n=z$ the identity:

$w=z^{1/n}=r^{1/n}\left(cos\left(\dfrac{\theta +2\pi k}{n}\right)+i\sin\left(\dfrac{\theta +2\pi k}{n}\right)\right)\forall k=0,1,\ldots,n-1$

The complex logarithm (or the complex power) is a multivalued functions (be aware!):

$\ln( re^{i\theta})=\ln r +i\theta +2\pi k,\forall k\in \mathbb{Z}$

The introduction of complex numbers and complex values of trigonometric functions are fun. You can check that

$\cos z=\dfrac{\exp (iz)+\exp (-iz)}{2}$

and

$\sin z=\dfrac{\exp (iz)-\exp (-iz)}{2i}$

and

$\tan z=\dfrac{\exp (iz)-\exp (-iz)}{i(\exp (iz)+\exp(-iz))}$

thanks to the Euler identity.

In special relativity, the geometry is “hyperbolic”, i.e., it is non-euclidean. Let me review the so-called hyperbolic trigonometry. More precisely, we are going to review the hyperbolic functions related to special relativity now.

HYPERBOLIC TRIGONOMETRY

We define the functions sinh, cosh and tanh ( sometimes written as sh, ch, th):

$\sinh x=\dfrac{\exp (x) -\exp (-x)}{2}$

$\cosh x=\dfrac{\exp (x) +\exp (-x)}{2}$

$\tanh x=\dfrac{\exp (x) -\exp (-x)}{\exp (x)+\exp (-x)}$

The fundamental theorem of hyperbolic trigonometry is

$\cosh^2 x-\sinh^2 x=1$

The hyperbolic triangles are objects like this:

The hyperbolic inverse functions are

$\sinh^{-1} x=\ln (x+\sqrt{x^2+1})$

$\cosh^{-1} x=\ln (x+\sqrt{x^2-1})$

$\tanh^{-1} x=\dfrac{1}{2}\ln\dfrac{1+x}{1-x}$

Two specially useful formulae in Special Relativity (related to the gamma factor, the velocity and a parameter called rapidity) are:

$\boxed{\sinh \tanh^{-1} x=\dfrac{x}{\sqrt{1-x^2}}}$

$\boxed{\cosh \tanh^{-1} x=\dfrac{1}{\sqrt{1-x^2}}}$

In fact, we also have:

$\exp(x)=\sinh x+\cosh x$

$\exp(-x)=-\sinh x+\cosh x$

$\sec\mbox{h}^2x+\tanh^2 x=1$

$\coth ^2 x-\csc\mbox{h}^2 x=1$

There are even more identities to be known. The most remarkable and important are likely to be:

$\sinh (x\pm y)=\sinh (x)\cosh (y)\pm\sinh (y)\cosh (x)$

$\cosh (x\pm y)=\cosh (x)\cosh (y)\pm\sinh (x)\sinh (y)$

$\tanh (x\pm y)=\dfrac{\tanh x\pm \tanh y}{1\pm \tanh x\tanh y}$

$\coth (x\pm y)=\dfrac{\coth x\coth y\pm 1}{\coth y\pm \coth x}$

You can also relate euclidean trigonometric functions with hyperbolic trigonometric functions with the aid of complex numbers. For instance, we get

$\sinh x=-i\sin ix$

$\cosh x=\cos ix$

$\tanh x= -i\tan ix$

and so on. The hyperbolic models of geometry/trigonometry are also very known in arts. Escher’s drawings are very beautiful and famous:

or the colorful variation of this theme

I love Escher’s drawings. And I also love Mathematics, Physics, Physmatics, and Science. Equations are cool. And hyperbolic functions, and other functions we have reviewed here today, will arise naturally in the next posts.

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Comments

LOG#023. Math survey. — 1 Comment

Zan (@Pythagorean_Ian) on 2012/12/06 at 5:02 AM said:

It’s too bad it’s too late to give $\pi/4$ eminence over $\pi$ , given that $\pi/4$ is the sum of the infinite series
$\displaystyle{\pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 -\ldots=\sum_{n=0}^\infty \dfrac{(-1)^n}{2n+1}}$

It’s , thus, the ratio of the circumference of a circle to the perimeter of a containing square, and it’s the ratio of the area of a circle to the area of a containing square. It is the glue between the curved and the linear.

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LOG#023. Math survey.

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