# Differential Equation in Spherical Harmonics Derivation

I've been reviewing derivations of the spherical harmonics in quantum mechanics; mostly as review but also to make sure I understand where the concepts arise from.

However, every derivation I've seen makes use of the following differential equation/ identity that seemingly comes from nowhere. I've not yet been able to figure out where it comes from in looking through various texts. So, here I am to ask: where does this come from?

$$\frac{d}{d\theta}+l cot(\theta) \equiv \frac{1}{(sin(\theta))^l} \frac{d}{d\theta}(sin(\theta)^l)$$

-

## 1 Answer

It comes from various differentiation rules. It is probably easier to grasp if one applies both sides of the identity on some function $f(\theta)$. Then the identity becomes

$$\frac{1}{\sin^\ell\theta} \frac{d}{d\theta}(\sin^\ell\theta~ f(\theta))~=~ f^\prime(\theta)+\ell \cot\theta ~f(\theta).$$

-

Ah I see it now, thanks! – atomicpedals Nov 19 '11 at 22:23
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9663516879081726, "perplexity_flag": "middle"}