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Sariah Hammond
Mathematics
13 May, 08:33
For which value of e is sin e=-1
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Annie Villanueva
13 May, 08:59
0
=ei x
Step-by-step explanation:
Consider the function on the right hand side (RHS)
f (x) = cos (x) + i sin (x)
Differentiate this function
f ' (x) = - sin (x) + i cos (x) = i f (x)
So, this function has the property that its derivative is i times the original function.
What other type of function has this property?
A function g (x) will have this property if
dg / dx = i g
This is a differential equation that can be solved with seperation of variables
(1/g) dg = i dx
integral (1/g) dg = integral i dx
ln| g | = i x + C
| g | = ei x + C = eC ei x
| g | = C2 ei x
g = C3 ei x
So we need to determine what value (if any) of the constant C3 makes g (x) = f (x).
If we set x=0 and evaluate f (x) and g (x), we get
f (x) = cos (0) + i sin (0) = 1
g (x) = C3 ei 0 = C3
These functions are equal when C3 = 1.
Therefore,
cos (x) + i sin (x) = ei x
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