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Complex
Analysis. FreeTutorial
7.1
Positive integer powers of i
7.2
Negative integer powers of i
7.3
Formulas for any integer powers of
i
7.1 Positive
integer powers of i
i2
= -1.
What is about i3,
i4,
i5,
im?
| i
0
= 1 |
| i
4
= 1 |
| i
8
= 1 |
| i
4m
= 1 |
|
| i
1
= i |
| i
5
= i |
| i
9
= i |
| i
4m+1
= i |
|
| i
2
= -1 |
| i
6
= -1 |
| i
10
= -1 |
| i
4m+2
= -1 |
|
| i
3
= -i |
| i
7
= -i |
| i
11
= -i |
| i
4m+3
= -i |
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where
m = 0,
1, 2, 3,
…
Positive powers of i
are periodic with period 4.
To evaluate i
m we have to replace
m with its remainder
on division by 4.
Example:
Evaluate i203.
We have to replace 203
with its remainder on division by 4.
203 = 4*50
+ 3;
i203
= i3
= – i.
Top
7.2
Negative integer powers of i
| i
-1
= -i |
| i
-5
= -i |
| i
-9
= -i |
| i
4m+3
= -i |
|
| i
-2
= -1 |
| i
-6
= -1 |
| i
-10
= -1 |
| i
4m+2
= -1 |
|
| i
-3
= i |
| i
-7
= i |
| i
-11
= i |
| i
4m+1
= i |
|
| i
-4
= 1 |
| i
-8
= 1 |
| i
-12
= 1 |
| i
4m
= 1 |
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where
m = -1,
-2, -3,
…
Negative powers of i
are periodic with period 4.
i
-1 = – i.
The reciprocal of i
is its own negation – i.
Top
7.3
Formulas for any integer powers of
i
Let us compare formulas
for positive and negative powers of i
to get formulas for any integer powers
of i.
i
4m =
1 |
i
4m+1
= i |
i
4m+2
= -1 |
i
4m+3
= -i |
where
m = 0,
±1,
±2,
±3,
…
Integer powers of i
are periodic with period 4.
To evaluate i
m we have to replace
m with its remainder
on division by 4.
by
Tetyana Butler
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