Formal Development of complex number

Http://en.wikipedia.org/wiki/Complex_number#Formal_development

In a rigorous setting, it is not acceptable to simply assume that there exists a number whose square is-1. the definition must therefore be a little less intuitive, building on the knowledge of real numbers. writeCForR2, the set of ordered pairs of real numbers, and define operations on complex numbers inCAccording

( A, B) + ( C, D) = ( A+ C, B+ D)

( A, B)·( C, D) = ( A. c− B · d, B. C+ A. D)

Since (0, 1) · (0, 1) = (−1, 0), we have foundIBy constructing it, not postulating it. We can associate the numbers (A, 0) with the real numbers, and writeI= (0, 1). It is then just a matter of notation to express (A,B)A+IB.

 

 

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Http://en.wikipedia.org/wiki/Euler%27s_formula

Definitions of complex exponentiation

Main Articles: Exponentiation and exponential function

In general, raisingETo a positive integer exponent has a simple interpretation in terms of repeated multiplicationE. RaisingETo zero or a negative integer exponent can be understood as repeated division. A rational number exponent can be defined by radicalsE, And an irrational number exponent can be defined by finding rational-number exponents that are arbitrarily close to the irrational-number exponent, In a limit process. however, to define and understand a complex number exponentE, A different type of generalization is required for the concept of exponentiation.

In fact, several definitions are possible. All of them can be proven to be well-defined and equivalent, although the proofs are not supported ded in this article.

Taylor series Definition

It is well-known that, for any realX, The following series is equalEX:

(In other words, this is the Taylor series for the real exponential function, and it has an infinite radius of convergence). This invites the followingDefinitionOfEZFor complexZ:

This can be proven to be well-defined; In particle, the series converges for anyZ.

Analytic continuation Definition

A simple-to-State, equivalent definition is thatEZ, For complexZ, Is the analytic continuation of the FunctionEXFor realX. This can be proven to be well-defined; In particle, it yields a single-valued function on the complex plane.

Limit Definition

It is well-known that, for any realX, The following limit is equalEX:

This motivates the followingDefinitionOfEZFor complexZ:

Differential Equation Definition

For realX, The FunctionF (x) = exIs well-known to be the unique real function satisfying the differential equation:

For allX. This motivates a definitionF (z) = EZFor complexZAs the function that satisfies the differential equation:

For all complexZ, Where the derivative inF'(Z) Is defined in the sense of a complex derivative. This can be proven to yield a unique function which is well-defined everywhere on the complex plane.

Proofs

Various proofs of this formula are possible. The first proof Below starts with the "Taylor series definition"EZ, While the other two use the "differential equation definition"EZ(See above ).

Using Taylor series

Here is a proof of Euler's formula using Taylor series expansions as well as basic facts about the powersI:

I ^ 0 & {}= 1, \ quad &
I ^ 1 & {}= I, \ quad &
I ^ 2 & {}=-1, \ quad &
I ^ 3 & {}=-I ,\\
I ^ 4 & ={} 1, \ quad &
I ^ 5 & ={} I, \ quad &
I ^ 6 & {}=-1, \ quad &
I ^ 7 & {}=-I ,\\
\ End {Align} "class =" Tex ">

And so on. The functionsEX, CosXAnd sinXOf the (real) VariableXCan be expressed using their Taylor expansions around zero:

E ^ X & {}= 1 + x + \ frac {x ^ 2} {2 !} + \ Frac {x ^ 3} {3 !} + \ Cdots \\
\ Cos x & {}= 1-\ frac {x ^ 2} {2 !} + \ Frac {x ^ 4} {4 !} -\ Frac {x ^ 6} {6 !} + \ Cdots \\
\ SiN x & {}= X-\ frac {x ^ 3} {3 !} + \ Frac {x ^ 5} {5 !} -\ Frac {x ^ 7} {7 !} + \ Cdots.
\ End {Align} "class =" Tex ">

For complexZWeDefineEach of these functions by the above series, replacing the Real VariableXWith the complex variableZ. This is possible because the radius of convergence of each series is infinite. We then find that

E ^ {iz} & {}= 1 + iz + \ frac {(IZ) ^ 2} {2 !} + \ Frac {(IZ) ^ 3} {3 !} + \ Frac {(IZ) ^ 4} {4 !} + \ Frac {(IZ) ^ 5} {5 !} + \ Frac {(IZ) ^ 6} {6 !} + \ Frac {(IZ) ^ 7} {7 !} + \ Frac {(IZ) ^ 8} {8 !} + \ Cdots \\
& {}= 1 + Iz-\ frac {z ^ 2} {2 !} -\ Frac {iz ^ 3} {3 !} + \ Frac {z ^ 4} {4 !} + \ Frac {iz ^ 5} {5 !} -\ Frac {z ^ 6} {6 !} -\ Frac {iz ^ 7} {7 !} + \ Frac {z ^ 8} {8 !} + \ Cdots \\
& {}=\ Left (1-\ frac {z ^ 2} {2 !} + \ Frac {z ^ 4} {4 !} -\ Frac {z ^ 6} {6 !} + \ Frac {z ^ 8} {8 !} -\ Cdots \ right) + I \ left (Z-\ frac {z ^ 3} {3 !} + \ Frac {z ^ 5} {5 !} -\ Frac {z ^ 7} {7 !} + \ Cdots \ right )\\
& {}= \ Cos Z + I \ sin Z
\ End {Align} "class =" Tex ">

The rearrangement of terms is justified because each series is absolutely convergent. TakingZ=XTo be a real number gives the original identity as Euler discovered it.

Q. e.d.

Using Calculus

Define the (possibly complex) function compute (X), Of Real VariableX,

The derivative of alternative (X), According to the product rule, is:

\ Frac {d} {DX} f (x) & {}= (\ Cos x + I \ SiN x) \ cdot \ frac {d} {DX} e ^ {-Ix} + \ frac {d} {DX} (\ Cos x + I \ SiN x) \ cdot e ^ {-Ix }\\
& {}= (\ Cos x + I \ SiN x) (-I e ^ {-Ix}) + (-\ SiN x + I \ Cos x) \ cdot e ^ {-Ix }\\
& {}= (-I \ Cos x-I ^ 2 \ SiN x) \ cdot e ^ {-Ix} + (-\ SiN x + I \ Cos x) \ cdot e ^ {-Ix} \ quad \ Quad (I ^ 2 =-1 )\\
& {}= (-I \ Cos x + \ SiN x-\ SiN x + I \ Cos x) \ cdot e ^ {-Ix }\\
& {} = 0.
\ End {Align} "class =" Tex ">

Therefore, equals (X) Must be a constant function inX. Because sums (0) is known, the constant that sums (X) Equals for all realXIs also known. Thus,

Multiplying both sidesEixAnd using

It follows that

Q. e.d.

Using Ordinary Differential Equations

Define the FunctionG(X)

Considering thatIIs constant, the First and Second DerivativesG(X) Are

BecauseI2 = −1 by definition. From this the following 2nd-order linear ordinary differential equation is constructed:

Or

Being a 2nd-order differential equation, there are two Linearly Independent solutions that satisfy it:

Both COs and sin are real functions in which the 2nd Derivative is identical to the negative of that function. any linear combination of solutions to a homogeneous differential equation is also a solution. then, in general, the solution to the differential equation is

For any constantsAAndB. But not all values of these two constants satisfy the known initial conditionsG(X):

.

However these same initial conditions (applied to the general solution) are

Resulting in

And, finally,