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
i^{2}
= 1.
What is about i^{3},
i^{4},
i^{5},
i^{m}?
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 

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 i^{203}.
We have to replace 203
with its remainder on division by 4.
203 = 4*50
+ 3;
i^{203}
= i^{3}
= – i.
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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 

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^{}.
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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
