## List of Complex Variables Formulas Latex Code

rockingdingo 2023-03-12 #Complex Variables #Conjugate #Power Series 2 0

List of Complex Variables Formulas Latex Code

In this blog, we will summarize the latex code for complex variables formulas, including complex numbers, De Moivreâ??s theorem and power series for complex variables e^{z}, sin(z), cos(z), ln(1+z), (1+z)^{n}, etc.

### 1. Complex Variables Formulas

• #### 1.0 Complex Numbers

##### Latex Code
            z=x+iy=r(\cos \theta + i \sin \theta)=re^{i(\theta+2n\pi)} \\
z^{*}=x-iy=r(\cos \theta - i \sin \theta)=re^{-i\theta} \\
zz^{*}=|z|^{2}=x^{2}+y^{2}

##### Explanation

• : i denotes the complex component of a complex number.
• : The real quantity r is the modulus of z.
• : The angle \theta is the argument of z.
• : The complex conjugate of z.

• #### 1.1 De Moivre's Theorem

##### Latex Code
            (\cos \theta + i \sin \theta)^{n}=e^{in\theta}=\cos n\theta + i \sin n\theta

##### Explanation

• : The power n of equals to

• #### 1.2 Power Series for Complex Variables

##### Latex Code
            e^{z}=1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+...+\frac{z^{n}}{n!}+...

##### Explanation

• : The z power e equals to summation of power series of

• #### 1.3 Power Series for Complex Variables sin(z)

##### Latex Code
            \sin z=z-\frac{z^{3}}{3!}+\frac{z^{5}}{5!}-...

##### Explanation

• : The sin(z) equals to summation of power series of

• #### 1.4 Power Series for Complex Variables cos(z)

##### Latex Code
            \cos z=1-\frac{z^{2}}{2!}+\frac{z^{4}}{4!}-...

##### Explanation

• : The cos(z) equals to summation of power series of

• #### 1.5 Power Series for Complex Variables ln(1+z)

##### Latex Code
            \ln (1+z)=1-\frac{z^{2}}{2!}+\frac{z^{3}}{3!}-...

##### Explanation

• : The ln(1+z) equals to summation of power series of

• #### 1.6 Power Series for Complex Variables (1+z)^{n}

##### Latex Code
            (1+z)^{n}=1+nz+\frac{n(n-1)}{2!}z^{2}+\frac{n(n-1)(n-2)}{3!}z^{3}+...

##### Explanation

• : The ln(1+z) equals to summation of power series of