In M2 we have learnt vectors, simple computational product problems (those proofings have left in the graveyard of HKALE), distance formula and etc. We tried a few problems on geometry which can be simply sovled by Euclidean Geometry tricks like parallel lines, etc. We simply don't know why we are going to proof it again with some stupid approach, right? Right now we will introduce some cool questions that can be solved by existing knowledge, but not by simple geometry.

I will not discuss lot about median and centroid because this relys mid-pt formula a lot, which is rather easy.

Example 1. Cosine law.

Consider triangle ABC,

## Wednesday, 22 February 2012

## Monday, 20 February 2012

### Visualization. 1

如果有那麼一個世界，以音樂本身作為存在的本質，那大概便是Osu!.

音樂，或說是一種波動，是慰藉心靈的方法。在這裡，音樂作為存在的本質，或是驅使元素的媒介。

在這個世界中，音樂被施以各種陣式來加以驅動；每演示一次成功的陣式都是對元素的召喚，與其說是召喚，不如當成加以凝聚去發出應有的能量。正如我們所認識的電子遍佈世界每一角落，但只有電壓才能驅使電子成為能源一樣，完成陣法的人簡單來說也就是操縱力量的人。

陣式，自然是古人創作出來的東西。一般人做出來的陣式雖然不是不能驅動，而是太難完成，這樣的話只會有內傷這一個下場。後來有人發明了在陣式中加入指示，使完成陣式的難度大減，自此又有人研究音樂與陣式配合與否的關鍵。自古以來有一群人，稱之為蝙蝠一族，擁有對各種陣式的詳細了解，亦有不可思議的強大力量；長久以來，便是他們統治著Osu中的子民，靠的便是陣法能否有效，取決於他們的彈指之間，沒有他們的認可，陣式再精美也是廢物一件。當然，他們也當中也有盡力為Osu子民服務的人，只是民間的陣式太多，蝙蝠一族的人太少了。後來他們發明了一種方法，將蝙蝠的力量複製予有能之士，當中供仳們選拔的便被委任為紫衣神官，這樣一來人們乾脆稱蝙蝠一族為紅衣：原因無他，人們避諱他們身上平時掩藏的蝙蝠翼，因為傳說中只有被他們殺掉的人才會看見那華麗但可怕的雙翼。

音樂，或說是一種波動，是慰藉心靈的方法。在這裡，音樂作為存在的本質，或是驅使元素的媒介。

在這個世界中，音樂被施以各種陣式來加以驅動；每演示一次成功的陣式都是對元素的召喚，與其說是召喚，不如當成加以凝聚去發出應有的能量。正如我們所認識的電子遍佈世界每一角落，但只有電壓才能驅使電子成為能源一樣，完成陣法的人簡單來說也就是操縱力量的人。

陣式，自然是古人創作出來的東西。一般人做出來的陣式雖然不是不能驅動，而是太難完成，這樣的話只會有內傷這一個下場。後來有人發明了在陣式中加入指示，使完成陣式的難度大減，自此又有人研究音樂與陣式配合與否的關鍵。自古以來有一群人，稱之為蝙蝠一族，擁有對各種陣式的詳細了解，亦有不可思議的強大力量；長久以來，便是他們統治著Osu中的子民，靠的便是陣法能否有效，取決於他們的彈指之間，沒有他們的認可，陣式再精美也是廢物一件。當然，他們也當中也有盡力為Osu子民服務的人，只是民間的陣式太多，蝙蝠一族的人太少了。後來他們發明了一種方法，將蝙蝠的力量複製予有能之士，當中供仳們選拔的便被委任為紫衣神官，這樣一來人們乾脆稱蝙蝠一族為紅衣：原因無他，人們避諱他們身上平時掩藏的蝙蝠翼，因為傳說中只有被他們殺掉的人才會看見那華麗但可怕的雙翼。

## Saturday, 11 February 2012

### Real identities involving complex numbers

This one requires some very basic college knowledge but is probably neglected by most students. With such deriving approach we can remember much less identities and realize more of the beauty of mathematics.

Part I - Nature of exponential function

In college level we first define the number e and the textbook will probably show that expansion of e directly, in better case we can get that expansion series with Taylor expansion, but that's not rigour enough. A more advanced and appropiate way is to show the expansion series of e itself, is exponential.

Definition. is defined as the exponential function. Note that z is used instead of x because we want complex number involving here.

Lemma.

Since we are not assuming as exponential, we shall demonstrates its properties. Note that is trivial in the expansion series, we have by product rule, i.e., is constant, and when z=0 the constant is exp(c). Now set z=a, c=a+b gives the expected result.

There could be more properties to be shown, but that's enough for us to state it as exponential.

Definition. and .

Now , so

Substitute them into the expansion series gives

and .

And again, it automatically fits the Euler Identity:

, , and . Following with the angle sum identites, showing that the function defined is the trigonometric function. (Note: the function equation concerning angle sum formula has unique solution of trigonometric function as above, this is beyond our discussion.

Part II - deriving trigonometric identities

This part is already included in my BAS note, but I'd want to show this one again.

Theorem.

Proof: Equating the expansion of

Equating real and imaginary part,

.

Theorem.

Proof.

There're another trick dealing with trigonometric functions at higher power.

Example. Evaluate

In most textbook we will try to use reduction formula, or transform it into forms of but we can also change it into trigoonometric function of degree 1 to deal with it.

Let , then . Similarly, , .

Observe that and

Now

By such a transformation the integration could be done easily. Note that .

Part III - transformation of complex number into polar (cis) form

Definition. Polar form of complex number is in form of where The second part of the statement is called De Moivre's theorem.

The proof is simple considering the multiplication in forms of , and by angle sum formula.

Example. Turn into polar form.

Solution.

Example. Turn into polar form.

Solution.

.

Part IV - analogy on real polynomial identities

Example. Show that .

In fact this is a lemma from the theorem that "any prime can be resolved into sum of two perfect squares."

It can be done by direct expansion, but we can make it by complex number:

Let ,

, done.

Example. Show that

Note that (n=0,1,2,3, n=0 -> 1 roots) are the roots of , summing the roots gives (the imaginary parts eliminates each other since ), so we get the desired result.

More to explore:

I) Expand tangent and cotangent in terms of polynomials.

II) Express and by trigonometric functions of the first degree with proof by complex numbers.

III) Turn into polar form, and show that

IV1) Evaluate

IV2) In fact we have another real identities concerning sum of four squares:

which can be proved by

Reference

Walter Rudin. Real and complex analysis. (In fact, the analysis on exponential series is commonly appearing in most books)

Margaret M.Gow. A Course in pure mathematics.

Part I - Nature of exponential function

In college level we first define the number e and the textbook will probably show that expansion of e directly, in better case we can get that expansion series with Taylor expansion, but that's not rigour enough. A more advanced and appropiate way is to show the expansion series of e itself, is exponential.

Definition. is defined as the exponential function. Note that z is used instead of x because we want complex number involving here.

Lemma.

Since we are not assuming as exponential, we shall demonstrates its properties. Note that is trivial in the expansion series, we have by product rule, i.e., is constant, and when z=0 the constant is exp(c). Now set z=a, c=a+b gives the expected result.

There could be more properties to be shown, but that's enough for us to state it as exponential.

**Bridge between exponential function and trigonometric functions**Definition. and .

Now , so

Substitute them into the expansion series gives

and .

And again, it automatically fits the Euler Identity:

, , and . Following with the angle sum identites, showing that the function defined is the trigonometric function. (Note: the function equation concerning angle sum formula has unique solution of trigonometric function as above, this is beyond our discussion.

Part II - deriving trigonometric identities

This part is already included in my BAS note, but I'd want to show this one again.

Theorem.

Proof: Equating the expansion of

Equating real and imaginary part,

.

Theorem.

Proof.

There're another trick dealing with trigonometric functions at higher power.

Example. Evaluate

In most textbook we will try to use reduction formula, or transform it into forms of but we can also change it into trigoonometric function of degree 1 to deal with it.

Let , then . Similarly, , .

Observe that and

Now

By such a transformation the integration could be done easily. Note that .

Part III - transformation of complex number into polar (cis) form

Definition. Polar form of complex number is in form of where The second part of the statement is called De Moivre's theorem.

The proof is simple considering the multiplication in forms of , and by angle sum formula.

Example. Turn into polar form.

Solution.

Example. Turn into polar form.

Solution.

.

Part IV - analogy on real polynomial identities

Example. Show that .

In fact this is a lemma from the theorem that "any prime can be resolved into sum of two perfect squares."

It can be done by direct expansion, but we can make it by complex number:

Let ,

, done.

Example. Show that

Note that (n=0,1,2,3, n=0 -> 1 roots) are the roots of , summing the roots gives (the imaginary parts eliminates each other since ), so we get the desired result.

More to explore:

I) Expand tangent and cotangent in terms of polynomials.

II) Express and by trigonometric functions of the first degree with proof by complex numbers.

III) Turn into polar form, and show that

IV1) Evaluate

IV2) In fact we have another real identities concerning sum of four squares:

which can be proved by

*quaternion numbers*. Search for quaternion numbers and prove the above identity by quaternion numbers.Reference

Walter Rudin. Real and complex analysis. (In fact, the analysis on exponential series is commonly appearing in most books)

Margaret M.Gow. A Course in pure mathematics.

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