Why do the limits of the following two equations have opposite signs? (MATLAB R2024a Update 1 _20240418)

Why limit((1-(-1)^n)/n, n, 0) = -pi*1i ? limit((1-exp(-pi*n*1i))/n, n, 0) = pi*1i

limit((1-(-1i)^n)/n, n, 0) = limit((1-exp(-pi/2*n*1i))/n, n, 0) = (pi*1i)/2

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>> syms n theta

>> assume(n,"integer")

>> limit((1-(-1)^n)/n, n, 0)

ans =

-pi*1i

>> limit((1-exp(-pi*n*1i))/n, n, 0)

ans =

pi*1i

>> n_value = -10:10;

>> subs(exp(-pi*n*1i) - (-1).^n, n, n_value)

ans =

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

>> limit((1-(-1i)^n)/n, n, 0)

ans =

(pi*1i)/2

>> limit((1-exp(-pi/2*n*1i))/n, n, 0)

ans =

(pi*1i)/2

>> subs(exp(-pi/2*n*1i) - (-1i).^n, n, n_value)

ans =

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

>>

[Walter Roberson]

(running the code here)

syms n theta
assume(n,"integer")
limit((1-(-1)^n)/n, n, 0)

ans = 

limit((1-exp(-pi*n*1i))/n, n, 0)

ans = 

n_value = -10:10;
subs(exp(-pi*n*1i) - (-1).^n, n, n_value)

ans = 

limit((1-(-1i)^n)/n, n, 0)

ans = 

limit((1-exp(-pi/2*n*1i))/n, n, 0)

ans = 

subs(exp(-pi/2*n*1i) - (-1i).^n, n, n_value)

ans = 

[dequan wu]

limit((1-(-1)^n)/n, n, 0) have two results, and Matlab only gets one of them.

(1-(-1)^n) = (1+(1i)^n)*(1-(1i)^n)

or

(1-(-1)^n) = (1+(-1i)^n)*(1-(-1i)^n)

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>> limit((1+(-1i)^n), n, 0)

ans =

2

>> limit((1+(1i)^n), n, 0)

ans =

2

>> limit((1-(1i)^n)/n, n, 0)

ans =

-(pi*1i)/2

>> limit((1+(1i)^n)*(1-(1i)^n)/n, n, 0)

ans =

-pi*1i

>> limit((1+(-1i)^n)*(1-(-1i)^n)/n, n, 0)

ans =

pi*1i

>>

Answer 1

[David Goodmanson]

Hello dq,

I think the behaviour is reasonably clear. The first example is

limit((1-(-1)^n)/n, n, 0) = -pi*i.

If you look at log(-1) the answer is i*pi, because for complex quantities z = r*exp(i*theta), Matlab takes the angle as

-pi < theta <= pi.  So (-1) = exp(i*pi)

and the result follows .In the second example,

limit((1-exp(-pi*n*i))/n, n, 0) = pi*i

you have forced (-1) = exp(-i*pi) so the result has a sign change. The other two examples follow since

(-i) = exp(-i*pi/2).

In all cases,

lim[(1-exp(i*a)^n)/n] = lim[(1-exp(i*a*n))/n] = lim[(1-(1+i*a*n +O(n^2))/n] = -i*a      

[dequan wu]

Thanks！

exp(i*θ)^n = exp(n*(ln|exp(i*θ)| + i*(arg(exp(i*θ)) + 2*k*pi)))

= exp(n*(0+ i*(θ + 2*k*pi))), -pi < θ <= pi, k is integer.

Principal Value: = exp(i*θ*n)