Complex Number2

Complex Number2

Exponential Values of Sine and Cosine:

We have

Putting , we have

Similarly, Thus we have , Also by addition and subtraction , we have

and

and for complex number

When , Corrolary: Since , where is real or n complex , we have Thus , De Moivre’s theorem holds good , whether is

real or complexLogarithm of a Complex number:Let be a non-zero complex number. Then there always exists a complex number such that . is said to be logarithm of Again , where is an integerThis shows that if is a logarithm of , then

is also a logarithm of . This means that “logarithm of ” is a many-valued function of . This is denoted by Log Of the many values of logarithm of , a particularone is called the principal value and is denoted by

Expression of logarithm of a complex number: Since is a non-zero complex number, has a polar representation.

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Let ( a polar form with amp ) Let be a logarithm of . Then . This gives Therefore We have and These determine and , where is an integerTherefore i.e., LogThe principal value of Log , denoted by log , is thevalue corresponding to .Therefore logLogarithm of a negative real number:Logarithm of a negative real number is given byLog =Log =Log = where isan integer.And Thus Log = =Definition of :If be a non-zero complex number and be any

complex number, is defined by Log Since Log is many-valued, is a many-valued

function. The principal value of corresponds to the principal value of Log

Theorem: If be two distinct complex numbers suchthat , thenLog +Log =LogProof: Since , Otherwise Log , Log and Log are not defined Let Then Now, Log , where is an integer Log ,where is an integer

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Complex Number2

Log , where is an integer Log Log

, where

Since are arbitrary integers , Log +Log =LogNote: is not necessarily equal to . For example , let .

Then , , , ,

,

Now,

Theorem: If be distinct complex numbers such that

, then

Log Log =Log .

Proof: Since ,

Otherwise Log , Log and Log are not defined

Let

Now, Log , where is an integer Log ,where is an integer

Log , where is an integer

Log Log

, where

Since are arbitrary integers,

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Log Log Log

Note: is not necessarily equal to .

For example , let

Then , , , ,

,

Now,

Theorem: If and be a positive integer, LogLog

.☺ Let . Then Now, Log , where is an integer Log , where is an integer Log

, where Since is arbitrary and is a multiple of , each value of Log is a value of Log but not conversely. Therefore the set of values of Log is a proper subset of values of Log Log LogTheorem: If and are complex numbers where ,

but (the p.v. of )(the p.v. of ) = the p.v. of .☺ Log

, where is an integer , where is an integer

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Complex Number2

, where is an integerNow, Where are arbitrary integers , the set of values

is a subset of the set of values butnot conversely But the p.v. of the p.v. of the p.v. of (the p.v. of )(the p.v. of ) = the p.v. of

Theorem: If are complex numbers and , , but the principal value of (the

p.v. of ( the p.v. of .☺ Log , Log , Log

Now, Log Log Log Log Log

The p.v. of , the p.v. of and the p.v. of But ,in general p. v. of (the p.v. of ( the p.v. of

Theorem: When is a complex number Theorem: If are complex numbers

Proof:

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, where

, where

Theorem: If are real, Note:

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Since increases steadily with , it follows that the functions and are not bounded inabsolute value. But if be real, the functions and are bounded in absolute value, as and

are never greater than 1.Theorem: When is a complex number, ,

, Hyperbolic functions: When is real, the hyperbolic functions are defined by

,

.Properties:1. 2. 3. for all real 4. 5. When is a complex number, 6. When is a complex number, .Thoerem: If be complex numbers

Proof:

, where

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, where

Note: and are periodic function with period , is a periodic function of period . and are periodic function of period

, is a periodic function of period .Worked out exercises:▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬Exercise1: Express Log , , in the form

where are real and find . ☺ Since is a non-zero complex number, it has

a polar representation. Let ,

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Complex Number2

Then and ( Principal argument ) Let LogThen , or, This gives

We have Since , we have Therefore , where is an integerHence, Log = =

where

The principal value of Log corresponds to (since is the principal argument )Hence

Exercise2: Find Log and when , .

.☺ Let , , since and

These determine and so is not the principal argument

of

Log , where is an integerand

Exercise3: prove that sin

.☺ First note that , because otherwise is not defined.Let .

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, where .

Then and .

Now and .

Therefore, .

Now .

Therefore .

Exercise4: Find the value of . .☺ Log

Sine , Log

Note: The values of are all real.Exercise5: Find the principal value of

.☺ Since , i.e.,

Therefore the p.v. of

Exercise6: Find all complex numbers such that

.☺ Let Then implies , and ,

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Now, , and , where is an integer .

Another method: . implies Log = , where is an

integer.Hence , where is an integer. Exercise7: Find Log .☺

, where is aninteger. Log

, where is an integer. .

Exercise8: Express in the form where are real . ☺ Let

or, , i.e. , in ,

This determines and

Exercise9: Find the general and principal value of.

.☺ Log ………..(i)

Log , where is an integer

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Complex Number2

So, from (i) we have

This gives the general value of

Exercise10: Show that

where ,.Evaluate the ratio when

.☺ Since ,

Log , where is an integerSimilarly , Log , where is an integer The principal value of is equal to

and value of the is equal to

2nd part: when ,

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Now Therefore

=The p.v. of

Exercise11: Show that

.☺ Log ,where is an integer

Log ,where is an integer

Log

Log

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Exercise12: Show that the values of can be arranged sothat they form a G.P.

. ☺ . Therefore, Log where isan integer Log where is an integerThe values of are ……., ,……. This isa G.P. with common ratio . Exercise13: Find the general values of , where , , and are realIf show that the points represented by them lie on the equiangular spiral in the complex plane. ☺ Since Log ,where is an

integer. ……..(1) Let …………(2)

Log

, [ using (1) ]

……(3)This gives the general values of From (2) and (3) we have ,

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………………(4) ……………….(5)

form (4) and (5) we have, or, ……………….(6)

and

…………...(7)

Using (6) and (7) we get,

.

.

Locus of is ,

Thus, the points represented by lie

on the equiangular spiral Exercise14: Find all values of such that .☺ Let

Then …………..(i) and ……………(ii)From (i) since . , where is an integer

From (ii) , since , where is an integer.

Exercise15: Find the general solution of .☺

or, , where

or, ,

or,

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When , =Log where is an integer When , =Log where is an integer

, where is an integerExercise16: Find the general solution of .☺

or, , where

or,

or, ,

When , =Log

Since,

Therefore Log = , where is

an integer

=

When , =Log

Since,

Therefore Log = , where is

an integer

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=

where is an integer

Exercise17: Find the general solution of

. ☺ or, , where

or, Therefore When , =LogSince Therefore, , where is an integerWhen , =LogSince Therefore, , where is an integerCombining we have, where is aninteger Exercise18: If where are

real and , show that and

.☺or,

or,

or,

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or,

or,

,

Squaring and adding, or, , since

or,

Also and

where is an integer

or,

Exercise19: If where are real , prove that (i) (ii) .☺

Let , where Then and therefore Now ,

Now,

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or, or,

and ,

or,

or,

or,

or,

Exercise20: If ,where is real , prove

that Log

.☺ Since is real , is real

or,

or, , where .

or,

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Complex Number2

or,

or, Log

or, Log

Exercise21: Find . ☺ Let . Then .

gives We have

When Log

Since ,

Therefore, Log = , where is

an integer.

So that

When Log

Since ,

Therefore, Log = , where is

an integer.So that

Therefore, , where is an

integer.

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and

Exercise22: Find .. ☺ Let . Then .

gives We have

When Log

Since ,

Therefore, Log = , where is an

integer.So that

When Log

Since ,


integer.So that =

Therefore, , where is an

integer. and

Exercise23: Find such that ,

.☺

or, , where

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or, ,

or,

.

.

or, Log

, since .

where is an integer.

or,

Exercise24: If , where are real and , prove that (i) and (ii) . ☺ If , is not defined.

Now Therefore , since .Now

or, or,

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But

But

Therefore, or, Exercise25: If , where is a real number and , then prove that lieson an ellipse☺. Let . Then or, or, Therefore, or, , where is an integer Therefore or, or,

So that

Now,

or,

Since lies on an ellipse. Exrecise26: If , where are real , prove that and are the roots of the equation . ☺ Let

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Compairing real and imaginary part we have,

, Now, . and are the roots of or, .Exercise27: If , prove that

(i) ,

(ii) .☺ Let

Compairing real and imaginary parts , we have,,

, (given) , .

Exercise28: If , then prove that

. ☺ Since ,

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or,

or,

[ Since and ]

or,

or,

or,

Exercise29: If prove that , where constant , the point lies on a family of confocal hyperbolas; and when constant , the point

lies on a family of confocal ellipses , beinga parameter in both the cases.. ☺ =

Then and When =constant, ,

…….(1)

(1) represents a hyperbola The eccentricity of the hyperbola (1) is

The foci of the hyperbola (1) are

Thus when =constant the point lies on a family of confocal hyperbolas. When =constant, ,

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…….(2)

(2) represents an ellipseThe eccentricity of the ellipse (2) is

The foci of the ellipse (2) are Thus when =constant the point lies on a family of confocal ellipses.Exercise30: Show that Log Log

. ☺ and


integerand Log = , where is an integer. So that 2LogTherefore all the values of Log is one of the values of Log but not conversely. Hnece Log LogExercise31: If is positive real, prove that , where is an integer. . ☺ Let , where is a positive real number.

Then Log , where is an integer. Therefore . Exercise32: If is negative real, prove that

, where is an integer. . ☺ Let , where is a positive real

number. Then Log , where is an integer. Therefore . Exercise33: If where is a positive real number, then prove that , where is an integer.

. ☺ Let .

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Then Log , where is an integer.

Therefore , where is an integer. Exercise34: Find all values of such that (i) (ii)

, (iii) , (iv) . . ☺ (i) Since , Log , where is an integer.

Log , where is an integer.

(ii) Since , Log , where

is an integer.Log , where is an integer.

(iii) Since , Log ,

where is an integer.Log , where is an integer.

Therefore, , where is an integer.

(iv) Let where . Then and . Therefore , i.e., . Thus and . These imply that .

Now Log , where isan integer… ...(1)Therefore … … (2)Multiplying (1) by 2 and then subtracting (2) from (1) we have,

Or, , where is an integer.

Exercise35: Show that .

. ☺ Since , .

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Since , .

Now .

Therefore .

.

Exercise36: Show that .

. ☺ Since , .

Since , .

Now .

Therefore .

.

Exercise37: Show that Log Log but.

. ☺ Since , Log , where is an integer. Therefore, Log , where is an integer.… … (i)

Therefore Log , where

is an integer. … … … (ii )From (i) and (ii) it follows that Log Log .

Therefore

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.

Thus we have, . Exercise38: Find the values of Log , .

. ☺ Since we have,

Log , where is an

integer. .

Exercise39: Express in the form where and are real.

. ☺

Since , and therefore and lies

in .

Thus we have, . This is fo the form where and both are real. Exercise40: If where are all real,prove that either , or . . ☺ Let , where are all real.

Then we have Or, Compairing real and imaginary parts we have, …(i), …(ii) From (ii) we have, . This implies either

or . When then .

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If then from (i) we have, .If then from (i) we have, i.e., . When we have

From (i) we have,

Or, or, .

Exercise41: If and , show that

if

if

if .

. ☺

Let .

Then and .

Therefore, or, and

.

If then we have , because .

If then we have , because .

If then we have .

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Thus if

if

if . Q42. Exercises: ▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬1. Find the values of (i) Log (ii)Log2. Express the following complex number in the form

(i)

(ii)

(iii)

3. Find the values of (i) (ii) (iii) 4. Find the general values of where

, and is a non zero real number. Showthat the points representing them in the complex planeare collinear.5. Find the general solution of 6.If , then show that and

7. If , where are real , provethat and are the roots of the equation

8. If prove that 9. (i) If , prove that (ii) If , prove that .

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