What is expand e^{(x+iy)^2} ?

The solution to expand e^{(x+iy)^2} is e^{x^2-y^2}cos(2xy)+ie^{x^2-y^2}sin(2xy)

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expand e^{(x+iy)^2}

expand

e^{(x + i y)^{2}}

e^{x^{2} - y^{2}} cos (2 x y) + i e^{x^{2} - y^{2}} sin (2 x y)

Solution steps

e^{(x + i y)^{2}}

(x + i y)^{2} = x^{2} + 2 i x y - y^{2}

= e^{x^{2} + 2 i x y - y^{2}}

Apply imaginary number rule:

e^{a + ib} = e^{a} (cos (b) + i sin (b))

= e^{x^{2} - y^{2}} (cos (2 x y) + i sin (2 x y))

Apply the distributive law:

a (b + c) = ab + a c

a = e^{x^{2} - y^{2}}, b = cos (2 x y), c = sin (2 x y) i

= e^{x^{2} - y^{2}} cos (2 x y) + e^{x^{2} - y^{2}} sin (2 x y) i

= e^{x^{2} - y^{2}} cos (2 x y) + i e^{x^{2} - y^{2}} sin (2 x y)

(- e^{x} sin (x) + e^{x} cos (x))^{2}

\frac{1}{( 1 + x ) ^{4}}

\frac{5}{( x ^{2} + 4 ) ( x ^{2} - 1 )}

\frac{3 x ^{2}}{( x + 3 ) ^{2}}

\frac{( x - 1 ) ^{2}}{2}

What is expand e^{(x+iy)^2} ?
The solution to expand e^{(x+iy)^2} is e^{x^2-y^2}cos(2xy)+ie^{x^2-y^2}sin(2xy)