Complex Numbers Notes

1. Complex Numbers: Definition on the Basis of Ordered Pairs A complex number is an extension of real numbers and is expressed in the form: \[ z = (a, b) \] where \( a, b \in \mathbb{R} \). The number \( z \) is interpreted as: - \( a \): the real part , written as \( \Re(z) \) - \( b \): the imaginary part , written as \( \Im(z) \) Operations on Ordered Pairs For two complex numbers \( z_1 = (a, b) \), \( z_2 = (c, d) \): - Addition: \[ z_1 + z_2 = (a + c, b + d) \] - Multiplication: \[ z_1 \cdot z_2 = (ac - bd, ad + bc) \] This multiplication rule ensures that \( (0, 1)^2 = (-1, 0) \), which justifies the notation \( i = (0, 1) \) and \( i^2 = -1 \). Using this, we convert the ordered pair \( (a, b) \) into the standard form: \[ z = a + ib \] --- 2. Algebra of Complex Numbers Let \( z_1 = a + ib \), \( z_2 = c + id \), where \( a, b, c, d \in \mathbb{R} \). Basic Operations: - Addition : \[ z_1 + z_2 = (a + c) + i(b + d) \] - Subtraction: \[ z_1 - z_2 = (a - c) + i(b - d) \] - Multiplication: \[ z_1 z_2 = (ac - bd) + i(ad + bc) \] - Division (Multiply numerator and denominator by conjugate of denominator): \[ \frac{z_1}{z_2} = \frac{(a + ib)(c - id)}{c^2 + d^2} = \frac{(ac + bd) + i(bc - ad)}{c^2 + d^2} \] - Conjugate: \[ \overline{z} = a - ib \] - Properties: - \( z \cdot \overline{z} = |z|^2 \) - \( \overline{z_1 + z_2} = \overline{z_1} + \overline{z_2} \) - \( \overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2} \)
3. Modulus and Amplitude Modulus: The modulus of \( z = a + ib \) is the distance from the origin in the Argand plane: \[ |z| = \sqrt{a^2 + b^2} \] Amplitude (Argument): The argument or amplitude of \( z \) is the angle \( \theta \) made with the positive real axis: \[ \arg(z) = \theta = \tan^{-1} \left( \frac{b}{a} \right) \] Note: - It is multi-valued due to periodicity: \( \arg(z) = \theta + 2n\pi \), \( n \in \mathbb{Z} \) Principal value : \[ \text{Arg}(z) \in (-\pi, \pi] \] 4. Argand Diagram A complex number \( z = a + ib \) is represented by a point \( (a, b) \) in the 2D plane (Argand plane). - The horizontal axis represents the real part. - The vertical axis represents the imaginary part. - The modulus is the length of the vector from the origin to the point. - The argument is the angle the vector makes with the real axis. This allows geometric interpretation of operations like addition (vector addition), multiplication (rotation + scaling), etc.
5. De Moivre's Theorem If \[ z = r(\cos \theta + i \sin \theta) \], then for any integer n: \[ z^n = r^n \left[ \cos(n\theta) + i \sin(n\theta) \right] \] Applications: - Roots of complex numbers : \[ z^{1/n} = r^{1/n} \left[ \cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right) \right], \quad k = 0, 1, \ldots, n-1 \] Expressing trigonometric identities by expanding \[ (\cos \theta + i \sin \theta)^n \] 6. Exponential, Sine, Cosine, and Logarithm of Complex Numbers Let \[ z = x + iy \]. Exponential Function : \[ e^z = e^{x+iy} = e^x (\cos y + i \sin y) \] This links complex exponentials with trigonometry: Euler's formula \[ e^{i\theta} = \cos \theta + i \sin \theta \] Sine and Cosine : - \[ \sin z = \frac{e^{iz} - e^{-iz}}{2i} \] - \[ \cos z = \frac{e^{iz} + e^{-iz}}{2} \] Logarithm of a Complex Number : For \( z = re^{i\theta} \), \[ \log z = \ln r + i(\theta + 2n\pi),\quad n \in \mathbb{Z} \] The logarithm is multi-valued due to the \( 2\pi \) periodicity of the angle. --- 7. Definition of \( a^x \) for \( a \ne 0 \) Let \[ a \in \mathbb{C},\ a \ne 0 \], and \[ x \in \mathbb{C} \]. Then: \[ a^x = e^{x \log a} \] Where \[ \log a \] is the complex logarithm of a, which may have multiple values depending on the branch.
8. Inverse Circular Functions Defined via logarithmic representations: - \[ \sin^{-1} z = -i \log \left( iz + \sqrt{1 - z^2} \right) \] - \[ \cos^{-1} z = -i \log \left( z + \sqrt{z^2 - 1} \right) \] - \[ \tan^{-1} z = \frac{i}{2} \log \left( \frac{1 - iz}{1 + iz} \right) \] These are multi-valued and defined using branch cuts in the complex plane. --- 9. Hyperbolic and Inverse Hyperbolic Functions Hyperbolic Functions : - \[ \sinh z = \frac{e^z - e^{-z}}{2} \] - \[ \cosh z = \frac{e^z + e^{-z}}{2} \] - \[ \tanh z = \frac{\sinh z}{\cosh z} \] Inverse Hyperbolic Functions : - \[ \sinh^{-1} z = \log \left( z + \sqrt{z^2 + 1} \right) \] - \[ \cosh^{-1} z = \log \left( z + \sqrt{z^2 - 1} \right) \] - \[ \tanh^{-1} z = \frac{1}{2} \log \left( \frac{1 + z}{1 - z} \right) \] These functions are also multi-valued due to the square root and logarithmic branches. ---