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Sagot :
Certainly! To expand [tex]\( f(x, y) = e^{xy} \)[/tex] as a Taylor series around the point [tex]\( (1, i) \)[/tex] up to and including the first degree terms, we need to follow these steps:
1. Evaluate the function at the point [tex]\( (1, i) \)[/tex]:
[tex]\[ f(1, i) = e^{1 \cdot i} = e^i \][/tex]
By Euler's formula, [tex]\( e^i = \cos(1) + i \sin(1) \)[/tex].
2. Compute the partial derivatives of [tex]\( f(x, y) \)[/tex]:
- First partial derivative with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{\partial f}{\partial x} = y e^{xy} \][/tex]
- Evaluating at [tex]\( (1, i) \)[/tex]:
[tex]\[ \left. \frac{\partial f}{\partial x} \right|_{(1, i)} = i e^i = i (\cos(1) + i \sin(1)) = i \cos(1) - \sin(1) \][/tex]
- First partial derivative with respect to [tex]\( y \)[/tex]:
[tex]\[ \frac{\partial f}{\partial y} = x e^{xy} \][/tex]
- Evaluating at [tex]\( (1, i) \)[/tex]:
[tex]\[ \left. \frac{\partial f}{\partial y} \right|_{(1, i)} = 1 \cdot e^i = e^i = \cos(1) + i \sin(1) \][/tex]
3. Construct the Taylor series expansion up to the first degree terms:
The Taylor series expansion of a function [tex]\( f(x, y) \)[/tex] around a point [tex]\( (a, b) \)[/tex] up to the first degree terms is given by:
[tex]\[ f(x, y) \approx f(a, b) + \left. \frac{\partial f}{\partial x} \right|_{(a, b)} (x - a) + \left. \frac{\partial f}{\partial y} \right|_{(a, b)} (y - b) \][/tex]
Substituting [tex]\( (a, b) = (1, i) \)[/tex], we get:
[tex]\[ f(x, y) \approx e^i + \left( i \cos(1) - \sin(1) \right)(x - 1) + \left( \cos(1) + i \sin(1) \right)(y - i) \][/tex]
4. Simplify the expression:
Combining all terms, the Taylor series expansion of [tex]\( e^{xy} \)[/tex] around [tex]\( (1, i) \)[/tex] up to the first degree is:
[tex]\[ e^{xy} \approx e^i + \left( i \cos(1) - \sin(1) \right)(x - 1) + \left( \cos(1) + i \sin(1) \right)(y - i) \][/tex]
By substituting the complex value [tex]\( e^i \)[/tex], [tex]\( i \cos(1) - \sin(1) \)[/tex], and [tex]\( \cos(1) + i \sin(1) \)[/tex] into the series, the expression remains straightforward yet combines the results yielding the final Taylor expansion around point [tex]\( (1, i) \)[/tex]:
[tex]\[ e^{xy} \approx (\cos(1) + i \sin(1)) + \left( i \cos(1) - \sin(1) \right)(x - 1) + \left( \cos(1) + i \sin(1) \right)(y - i) \][/tex]
This gives you a complete first-degree Taylor expansion of the function [tex]\( e^{xy} \)[/tex] around the point [tex]\( (1, i) \)[/tex].
1. Evaluate the function at the point [tex]\( (1, i) \)[/tex]:
[tex]\[ f(1, i) = e^{1 \cdot i} = e^i \][/tex]
By Euler's formula, [tex]\( e^i = \cos(1) + i \sin(1) \)[/tex].
2. Compute the partial derivatives of [tex]\( f(x, y) \)[/tex]:
- First partial derivative with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{\partial f}{\partial x} = y e^{xy} \][/tex]
- Evaluating at [tex]\( (1, i) \)[/tex]:
[tex]\[ \left. \frac{\partial f}{\partial x} \right|_{(1, i)} = i e^i = i (\cos(1) + i \sin(1)) = i \cos(1) - \sin(1) \][/tex]
- First partial derivative with respect to [tex]\( y \)[/tex]:
[tex]\[ \frac{\partial f}{\partial y} = x e^{xy} \][/tex]
- Evaluating at [tex]\( (1, i) \)[/tex]:
[tex]\[ \left. \frac{\partial f}{\partial y} \right|_{(1, i)} = 1 \cdot e^i = e^i = \cos(1) + i \sin(1) \][/tex]
3. Construct the Taylor series expansion up to the first degree terms:
The Taylor series expansion of a function [tex]\( f(x, y) \)[/tex] around a point [tex]\( (a, b) \)[/tex] up to the first degree terms is given by:
[tex]\[ f(x, y) \approx f(a, b) + \left. \frac{\partial f}{\partial x} \right|_{(a, b)} (x - a) + \left. \frac{\partial f}{\partial y} \right|_{(a, b)} (y - b) \][/tex]
Substituting [tex]\( (a, b) = (1, i) \)[/tex], we get:
[tex]\[ f(x, y) \approx e^i + \left( i \cos(1) - \sin(1) \right)(x - 1) + \left( \cos(1) + i \sin(1) \right)(y - i) \][/tex]
4. Simplify the expression:
Combining all terms, the Taylor series expansion of [tex]\( e^{xy} \)[/tex] around [tex]\( (1, i) \)[/tex] up to the first degree is:
[tex]\[ e^{xy} \approx e^i + \left( i \cos(1) - \sin(1) \right)(x - 1) + \left( \cos(1) + i \sin(1) \right)(y - i) \][/tex]
By substituting the complex value [tex]\( e^i \)[/tex], [tex]\( i \cos(1) - \sin(1) \)[/tex], and [tex]\( \cos(1) + i \sin(1) \)[/tex] into the series, the expression remains straightforward yet combines the results yielding the final Taylor expansion around point [tex]\( (1, i) \)[/tex]:
[tex]\[ e^{xy} \approx (\cos(1) + i \sin(1)) + \left( i \cos(1) - \sin(1) \right)(x - 1) + \left( \cos(1) + i \sin(1) \right)(y - i) \][/tex]
This gives you a complete first-degree Taylor expansion of the function [tex]\( e^{xy} \)[/tex] around the point [tex]\( (1, i) \)[/tex].
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